phanerozoic/Metamath · Datasets at Hugging Face

fact stringlengths 9 24k	type stringclasses 2 values	library stringclasses 5 values	imports listlengths 0 0	filename stringclasses 5 values	symbolic_name stringlengths 1 24	docstring stringlengths 12 292k ⌀
idi : \|- ph	theorem	set	[]	set.mm	idi	This is the Metamath database set.mm. $) $( Metamath is a formal language and associated computer program for archiving, verifying, and studying mathematical proofs, created by Norman Dwight Megill (1950--2021). For more information, visit https://us.metamath.org and https://github.com/metamath/set.mm, and feel free to ask questions at https://groups.google.com/g/metamath. $) $( New users may want to read https://us.metamath.org/mpeuni/conventions.html to understand the label naming conventions used in set.mm. See also the Metamath program command "MM> HELP VERIFY MARKUP" for markup conventions. $) $( To break this file into smaller modules, in the Metamath program type "MM> READ set.mm" followed by "MM> WRITE SOURCE set.mm / SPLIT". To recombine, omit "/ SPLIT". $) $( The database set.mm was created by Norman Megill on 30-Sep-1992 and has been continuously enriched since then (list of contributors below). $) $( ! ######################################## Metamath source file for logic and set theory ######################################## ~~ PUBLIC DOMAIN ~~ This work is waived of all rights, including copyright, according to the CC0 Public Domain Dedication. https://creativecommons.org/publicdomain/zero/1.0/ Currently active maintainers: See the list in the CONTRIBUTING.md file of https://github.com/metamath/set.mm. Contributor list: DA David Abernethy SA Stefan Allan TA Thierry Arnoux JA Juha Arpiainen JB Jonathan Ben-Naim GB Gregory Bush MC Mario Carneiro FC Filip Cernatescu PC Paul Chapman DF Drahflow AD Adrian Ducourtial GD Georgy Dunaev SF Scott Fenton GG Gino Giotto JGH Jeff Hankins AH Anthony Hart DH David Harvey CH Chen-Pang He JH Jeff Hoffman II Igor Ieskov AI Asger C. Ipsen JJ Jerry James SJ Szymon Jaroszewicz BJ Benoit Jubin JK Jim Kingdon ML M L WL Wolf Lammen GL Gerard Lang BL Brendan Leahy LL Larry Lesyna RL Raph Levien FL Frederic Line RFL Roy F. Longton TM T M JPM Jeffrey P. Machado JM Jeff Madsen GM Giovanni Mascellani PM Peter Mazsa RM Rodolfo Medina NM Norman Megill MKU metakunt DM David Moews MM Mykola Mostovenko SN Steven Nguyen MO Mel L. O'Cat OAI OpenAI SO Stefan O'Rear JO Jason Orendorff KP K P NP Noam Pasman JPP Jon Pennant RP Richard Penner SP Stanislas Polu JP Josh Purinton RMI Remi RR Rohan Ridenour SR Steve Rodriguez ATS Andrew Salmon AS Alan Sare ES Eric Schmidt GS Glauco Siliprandi SS Saveliy Skresanov BT BTernaryTau ET Ender Ting JU Jarvin Udandy ADH Stijn "Adhemar" Vandamme AV Alexander van der Vekens JV Jannik Vierling ZW Zhi Wang EW Emmett Weisz DAW David A. Wheeler RW Roger Witte KW Kyle Wyonch JY Jonathan Yan FZ Fan Zheng KZ Kunhao Zheng HTML code for accented names: BJ Benoît Jubin GL Gérard Lang FL Frédéric Liné $) $( See "MM> HELP VERIFY MARKUP" for help with modularization tags. $) $( Begin $[ set-header.mm $] $) $( ! =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Contents of this header =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= * Quick "How To" * Bibliography * Metamath syntax summary * Other notes =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Quick "How To" =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= How to use this file under Windows 95/98/NT/2K/XP/Vista/7/10: 1. Download the Metamath program metamath.exe following the instructions on the Metamath home page (https://us.metamath.org) and put it in the same directory as this file. 2. In Windows Explorer, double-click on metamath.exe. 3. Type "read set.mm" and press Enter. 4. Type "help" for a list of help topics, and "help demo" for some command examples. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Bibliography =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Bibliographical references are made by bracketing an identifier in a theorem's comment, such as [RussellWhitehead]. These refer to HTML tags on the following web pages: Logic and set theory - see https://us.metamath.org/mpeuni/mmset.html#bib Hilbert space - see https://us.metamath.org/mpeuni/mmhil.html#ref A bracketed reference must be preceded by a theorem number, etc. and followed by a page number. See "MM> HELP WRITE BIBLIOGRAPHY" for details. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Metamath syntax summary =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The HELP LANGUAGE command in the Metamath program will give you a quick overview of Metamath. The specification is found on pp. 111--114 of the Metamath book. The following syntax summary is provided for convenience but may omit some details. A Metamath database (set of one or more ASCII source files) is a sequence of _tokens_, which are normally separated by spaces or line breaks. The only tokens that are built into the Metamath language are those (two-character sequences) beginning with $, shown in the following. These tokens are called _keywords_: $c ... $. - Constant declaration $v ... $. - Variable declaration $d ... $. - Disjoint (distinct) variable restriction <label> $f ... $. - "Floating" hypothesis (i.e. variable type declaration) <label> $e ... $. - "Essential" hypothesis (i.e. a logical assumption for a theorem or axiom) <label> $a ... $. - Axiom or definition or syntax construction <label> $p ... $= ... $. - Theorem and its proof ${ ... $} - Block for defining the scope of the above statements (except $a, $p which are forever active) $) $( ... $) $( - Comments (may not be nested); see HELP LANGUAGE for markup features. $[ ... $] - Include a file The above two-character sequences beginning with "$" are the only primitives built into the Metamath language. The only "logic" Metamath uses in its proof verification algorithm is the substitution of expressions for variables while checking for distinct variable violations. Everything else, including the axioms for logic, is defined in this database file. All other tokens are user-defined, and their names are arbitrary. There are two kinds of user-defined tokens, called math symbols (or just symbols) and labels. A _symbol_ may contain any non-whitespace printable character except "$". A _label_ may contain only alphanumeric characters and the characters "." (period), "-" (hyphen), and "_" (underscore). Symbols and labels are case-sensitive. All labels (except in proofs) must be distinct. A label may not have the same name as a symbol (to simplify the coding of certain parsers and translators). Here is some more detail about the syntax: $c <symbollist> $. <symbollist> is a (whitespace-separated) list of distinct symbols that haven't been used before. $v <symbollist> $. <symbollist> is a list of distinct symbols that haven't been used yet in the current scope (see ${ ... $} below). $d <symbollist> $. <symbollist> is a (whitespace-separated) list of distinct symbols previously declared with $v in current scope. It means that substitutions into these symbols may not have variables in common. <label> $f <symbollist> $. <symbollist> is a list of 2 symbols, the first of which must be previously declared with $c in the current scope. <label> $e <symbollist> $. <symbollist> is a list of 2 or more symbols, the first of which must be previously declared with $c in the current scope. <label> $a <symbollist> $. <symbollist> is a list of 2 or more symbols, the first of which must be previously declared with $c in the current scope. <label> $p <symbollist> $= <proof> $. <symbollist> is a list of 2 or more symbols, the first of which must be previously declared with $c in the current scope. <proof> is either a whitespace-delimited sequence of previous labels (created by SAVE PROOF <label> /NORMAL) or a compressed proof (created by SAVE PROOF <label> /COMPRESSED). After using SAVE PROOF, use WRITE SOURCE to save the database file to disk. ${ ... $} Block for scoping the above statements (except $a, $p which are forever active). Currently, $c may not occur inside of a block. $) $( <any text> $) $( Comment. Note: <any text> may not contain adjacent "$" and ")" characters. The comment opening and closing delimiters must be surrounded by whitespace (space, tab, CR, LF, or FF). $[ <filename> $] Insert contents of <filename> at this point. If <filename> is current file or has been already been inserted, it will not be inserted again. Inside of comments, it is recommended that labels be preceded with a tilde (~) and math symbol tokens be enclosed in grave accents, also known as backticks (` `). These tildes, tokens, math symbols and backticks should be surrounded by spaces. This way the LaTeX and HTML rendition of comments will be accurate, and tools to globally change labels and math symbols will also change them in comments. Note that inside of backticks a pair of backticks is interpreted as a single backtick. A special comment containing $ t (with no space after the dollar sign) defines LaTeX and HTML symbols. See HELP LANGUAGE and HELP HTML for other markup features in comments. The proofs in this file are in "compressed" format for storage efficiency. The Metamath program reads the compressed format directly. This format is described in Appendix B of the Metamath book. It is not intended to be read by humans. For viewing proofs you should use the various SHOW PROOF commands described in the Metamath book (or the online HELP). The Metamath program does not normally affect any content of this file other than proofs, i.e., the text between "$=" and "$." (and some rewrapping). All other content is user-created. Proofs are created or modified with the PROVE command. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Other notes =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 1. It is recommended that you be familiar with Chapters 2 and 4 of the Metamath book to understand the Metamath language. Chapters 2, 3 and 5 explain how to use the Metamath program. Chapter 3 gives an informal overview of what this source file is all about. Appendix A gives the standard mathematical symbols corresponding to some of the ASCII tokens used in this file. The ASCII tokens may seem cryptic at first, even if you are familiar with set theory, but a review of the definition summary in Chapter 3 should quickly enable you to see the correspondence to standard mathematical notation. To easily find the definition of a token, search for the first occurrences of the token surrounded by spaces. Some odd-looking ones include "-." for "not", and "C_" for "is a subset of". The Metamath program "MM> HELP TEX" command explains how to obtain a LaTeX output to see the real mathematical symbols. Let us know if you have better suggestions for naming ASCII tokens. 2. Theorems can be written in different forms, including "closed form", "deduction form", and "inference form" (for details, see ~ conventions ). For basic theorems, all three forms are generally given, but for more advanced theorems, we prefer to use the deduction form, since it permits to write proofs in the "deduction style", and we do not add theorems in inference form unless there are reasonable grounds for it (for instance, shortening sufficiently many proofs to counterbalance their addition). 3. On providing new definitions and theorems, the conventions provided in the comment of ~ conventions should be obeyed. 4. For a chronological list of changes to label names and label deletions, see the changes-set.txt file. This should help if you have a proof not checked into the main repository and want to update it for recent changes. $) $( End $[ set-header.mm $] $) $( Begin $[ set-main.mm $] $) $( Begin $[ set-pred.mm $] $) $( The following header is the first to appear in the Theorem List contents, because higher-level headers suppress all previous same-level or lower-level headers in the same comment area between $a and $p statements. See "MM> HELP WRITE THEOREM_LIST" for information about headers. $) $( ############################################################################### CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY ############################################################################### Logic can be defined as the "study of the principles of correct reasoning" (Merrilee H. Salmon's 1991 "Informal Reasoning and Informal Logic" in _Informal Reasoning and Education_) or as "a formal system using symbolic techniques and mathematical methods to establish truth-values" (the Oxford English Dictionary). This section formally defines the logic system we will use. In particular, it defines symbols for declaring truthful statements, along with rules for deriving truthful statements from other truthful statements. The system defined here is classical first-order logic (often abbreviated as FOL) with equality and no terms (the most common logic system used by mathematicians). We begin with a few housekeeping items in pre-logic, and then introduce propositional calculus (both its axioms and important theorems that can be derived from them). Propositional calculus deals with general truths about well-formed formulas (wffs) regardless of how they are constructed. This is followed by proofs that other axiomatizations of classical propositional calculus can be derived from the axioms we have chosen to use. We then define predicate calculus, which adds additional symbols and rules useful for discussing objects (beyond simply true or false). In particular, it introduces the symbols ` = ` ("equals"), ` e. ` ("is a member of"), and ` A. ` ("for all"). The first two are called "predicates". A predicate specifies a true or false relationship between its two arguments. $) $( ######################################## Pre-logic ######################################## This section includes a few "housekeeping" mechanisms before we begin defining the basics of logic. $) $( Declare the primitive constant symbols for propositional calculus. $) $c ( $. $( Left parenthesis $) $c ) $. $( Right parenthesis $) $c -> $. $( Right arrow (read: "implies") $) $c -. $. $( Right handle (read: "not") $) $c wff $. $( Well-formed formula symbol (read: "the following symbol sequence is a wff") $) $c \|- $. $( Turnstile (read: "the following symbol sequence is provable" or "a proof exists for") $) $( Define the syntax and logical typecodes, and declare that our grammar is unambiguous (verifiable using the KLR parser, with compositing depth 5). (This $ j comment need not be read by verifiers, but is useful for parsers like mmj2.) $) $( $j syntax 'wff'; syntax '\|-' as 'wff'; unambiguous 'klr 5'; $) $( Declare the color of wff variables. $) $( $j varcolorcode "wff" as "0000FF"; altvarcolorcode "wff" as "337DFF"; $) $( Declare typographical constant symbols that are not directly used in the formalism but are useful to explain it in comments. $) $c & $. $( Ampersand (read: "and"). $) $c => $. $( Double right arrow (read: "implies"). $) $( wff variable sequence: ph ps ch th ta et ze si rh mu la ka $) $( Introduce some variable names we will use to represent well-formed formulas (wff's). $) $v ph $. $( Greek phi $) $v ps $. $( Greek psi $) $v ch $. $( Greek chi $) $v th $. $( Greek theta $) $v ta $. $( Greek tau $) $v et $. $( Greek eta $) $v ze $. $( Greek zeta $) $v si $. $( Greek sigma $) $v rh $. $( Greek rho $) $v mu $. $( Greek mu $) $v la $. $( Greek lambda $) $v ka $. $( Greek kappa $) $( Specify some variables that we will use to represent wff's. The fact that a variable represents a wff is relevant only to a theorem referring to that variable, so we may use $f hypotheses. The symbol ` wff ` specifies that the variable that follows it represents a wff. $) $( Let variable ` ph ` be a wff. $) wph $f wff ph $. $( Let variable ` ps ` be a wff. $) wps $f wff ps $. $( Let variable ` ch ` be a wff. $) wch $f wff ch $. $( Let variable ` th ` be a wff. $) wth $f wff th $. $( Let variable ` ta ` be a wff. $) wta $f wff ta $. $( Let variable ` et ` be a wff. $) wet $f wff et $. $( Let variable ` ze ` be a wff. $) wze $f wff ze $. $( Let variable ` si ` be a wff. $) wsi $f wff si $. $( Let variable ` rh ` be a wff. $) wrh $f wff rh $. $( Let variable ` mu ` be a wff. $) wmu $f wff mu $. $( Let variable ` la ` be a wff. $) wla $f wff la $. $( Let variable ` ka ` be a wff. $) wka $f wff ka $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Inferences for assisting proof development =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The inference rules in this section will normally never appear in a completed proof. They can be ignored if you are using this database to assist learning logic - please start with the statement ~ wn instead. $) ${ idi.1 $e \|- ph $. $( (_Note_: This inference rule and the next one, ~ a1ii , will normally never appear in a completed proof. They can be ignored if you are using this database to assist learning logic; please start with the statement ~ wn instead.) This inference says "if ` ph ` is true then ` ph ` is true". This inference requires no axioms for its proof, and is useful as a copy-paste mechanism during proof development in mmj2. It is normally not referenced in the final version of a proof, since it is always redundant. You can remove this using the metamath-exe (Metamath program) Proof Assistant using the "MM-PA> MINIMIZE__WITH *" command. This is the inference associated with ~ id , hence its name. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
a1ii : \|- ph	theorem	set	[]	set.mm	a1ii	(_Note_: This inference rule and the previous one, ~ idi , will normally never appear in a completed proof.) This is a technical inference to assist proof development. It provides a temporary way to add an independent subproof to a proof under development, for later assignment to a normal proof step. The Metamath (Metamath-exe) program Proof Assistant requires proofs to be developed backwards from the conclusion with no gaps, and it has no mechanism that lets the user work on isolated subproofs. This inference provides a workaround for this limitation. It can be inserted at any point in a proof to allow an independent subproof to be developed on the side, for later use as part of the final proof. _Instructions_: <HTML><ol><li>Assign this inference to any unknown step in the proof. Typically, the last unknown step is the most convenient, since <code>MM-PA> ASSIGN LAST</code> can be used. This step will be replicated in hypothesis a1ii.1, from where the development of the main proof can continue.</li><li>Develop the independent subproof backwards from hypothesis a1ii.2. If desired, use a <code>MM-PA> LET STEP</code> command to pre-assign the conclusion of the independent subproof to a1ii.2.</li><li>After the independent subproof is complete, use <code>MM-PA> IMPROVE ALL</code> to assign it automatically to an unknown step in the main proof that matches it.</li><li>After the entire proof is complete, use <code>MM-PA> MINIMIZE_WITH *</code> to clean up (discard) all ~ a1ii references automatically.</ol></HTML> This can also be used to apply subproofs from other theorems. In step 2, simply assign the theorem to a1ii.2, and run <HTML><code>MM-PA> EXPAND <theorem></code></HTML> to "import" a subproof from another theorem. This inference was originally designed to assist importing partially completed Proof Worksheets from the mmj2 Proof Assistant GUI, but it can also be useful on its own. Interestingly, no axioms are required for its proof. It is the inference associated with ~ a1i . (Contributed by NM, 7-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
mp2 : \|- ch	theorem	set	[]	set.mm	mp2	Register implication '->' as a primitive expression (lacking a definition). $) $( $j primitive 'wi'; $) $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The axioms of propositional calculus =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Propositional calculus (Axioms ~ ax-1 through ~ ax-3 and rule ~ ax-mp ) can be thought of as asserting formulas that are universally "true" when their variables are replaced by any combination of "true" and "false". Propositional calculus was first formalized by Frege in 1879, using as his axioms (in addition to rule ~ ax-mp ) the wffs ~ ax-1 , ~ ax-2 , ~ pm2.04 , ~ con3 , ~ notnot , and ~ notnotr . Around 1930, Lukasiewicz simplified the system by eliminating the third (which follows from the first two, as you can see by looking at the proof of ~ pm2.04 ) and replacing the last three with our ~ ax-3 . (Thanks to Ted Ulrich for this information.) The theorems of propositional calculus are also called _tautologies_. Tautologies can be proved very simply using truth tables, based on the true/false interpretation of propositional calculus. To do this, we assign all possible combinations of true and false to the wff variables and verify that the result (using the rules described in ~ wi and ~ wn ) always evaluates to true. This is called the _semantic_ approach. Our approach is called the _syntactic_ approach, in which everything is derived from axioms. A metatheorem called the Completeness Theorem for Propositional Calculus shows that the two approaches are equivalent and even provides an algorithm for automatically generating syntactic proofs from a truth table. Those proofs, however, tend to be long, since truth tables grow exponentially with the number of variables, and the much shorter proofs that we show here were found manually. $) ${ $( Minor premise for modus ponens. $) min $e \|- ph $. $( Major premise for modus ponens. $) maj $e \|- ( ph -> ps ) $. $( Rule of Modus Ponens. The postulated inference rule of propositional calculus. See, e.g., Rule 1 of [Hamilton] p. 73. The rule says, "if ` ph ` is true, and ` ph ` implies ` ps ` , then ` ps ` must also be true". This rule is sometimes called "detachment", since it detaches the minor premise from the major premise. "Modus ponens" is short for "modus ponendo ponens", a Latin phrase that means "the mode that by affirming affirms" - remark in [Sanford] p. 39. This rule is similar to the rule of modus tollens ~ mto . Note: In some web page displays such as the Statement List, the symbols " ` & ` " and " ` => ` " informally indicate the relationship between the hypotheses and the assertion (conclusion), abbreviating the English words "and" and "implies". They are not part of the formal language. (Contributed by NM, 30-Sep-1992.) $) ax-mp $a \|- ps $. $} $( Axiom _Simp_. Axiom A1 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. The 3 axioms are also given as Definition 2.1 of [Hamilton] p. 28. This axiom is called _Simp_ or "the principle of simplification" in _Principia Mathematica_ (Theorem 2.02 of [WhiteheadRussell] p. 100) because "it enables us to pass from the joint assertion of ` ph ` and ` ps ` to the assertion of ` ph ` simply". It is Proposition 1 of [Frege1879] p. 26, its first axiom. (Contributed by NM, 30-Sep-1992.) $) ax-1 $a \|- ( ph -> ( ps -> ph ) ) $. $( Axiom _Frege_. Axiom A2 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It "distributes" an antecedent over two consequents. This axiom was part of Frege's original system and is known as _Frege_ in the literature; see Proposition 2 of [Frege1879] p. 26. It is also proved as Theorem 2.77 of [WhiteheadRussell] p. 108. The other direction of this axiom also turns out to be true, as demonstrated by ~ pm5.41 . (Contributed by NM, 30-Sep-1992.) $) ax-2 $a \|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) $. $( Axiom _Transp_. Axiom A3 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky". This axiom is called _Transp_ or "the principle of transposition" in _Principia Mathematica_ (Theorem *2.17 of [WhiteheadRussell] p. 103). We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning. (Contributed by NM, 30-Sep-1992.) Use its alias ~ con4 instead. (New usage is discouraged.) $) ax-3 $a \|- ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Logical implication =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The results in this section are based on implication only, and avoid ~ ax-3 , so are intuitionistic. The system { ~ ax-mp , ~ ax-1 , ~ ax-2 } axiomatizes what is sometimes called "intuitionistic implicational calculus" or "minimal implicational calculus". In an implication, the wff before the arrow is called the "antecedent" and the wff after the arrow is called the "consequent". $) ${ mp2.1 $e \|- ph $. mp2.2 $e \|- ps $. mp2.3 $e \|- ( ph -> ( ps -> ch ) ) $. $( A double modus ponens inference. (Contributed by NM, 5-Apr-1994.)
mp2b : \|- ch	theorem	set	[]	set.mm	mp2b	A double modus ponens inference. (Contributed by Mario Carneiro, 24-Jan-2013.)
a1i : \|- ( ps -> ph )	theorem	set	[]	set.mm	a1i	null
2a1i : \|- ( ps -> ( ch -> ph ) )	theorem	set	[]	set.mm	2a1i	Inference introducing two antecedents. Two applications of ~ a1i . Inference associated with ~ 2a1 . (Contributed by Jeff Hankins, 4-Aug-2009.)
mp1i : \|- ( ch -> ps )	theorem	set	[]	set.mm	mp1i	Inference detaching an antecedent and introducing a new one. (Contributed by Stefan O'Rear, 29-Jan-2015.)
a2i : \|- ( ( ph -> ps ) -> ( ph -> ch ) )	theorem	set	[]	set.mm	a2i	Inference distributing an antecedent. Inference associated with ~ ax-2 . Its associated inference is ~ mpd . (Contributed by NM, 29-Dec-1992.)
mpd : \|- ( ph -> ch )	theorem	set	[]	set.mm	mpd	A modus ponens deduction. A translation of natural deduction rule ` -> ` E ( ` -> ` elimination), see ~ natded . Deduction form of ~ ax-mp . Inference associated with ~ a2i . Commuted form of ~ mpcom . (Contributed by NM, 29-Dec-1992.)
imim2i : \|- ( ( ch -> ph ) -> ( ch -> ps ) )	theorem	set	[]	set.mm	imim2i	null
syl : \|- ( ph -> ch )	theorem	set	[]	set.mm	syl	null
3syl : \|- ( ph -> th )	theorem	set	[]	set.mm	3syl	Inference chaining two syllogisms ~ syl . Inference associated with ~ imim12i . (Contributed by NM, 28-Dec-1992.)
4syl : \|- ( ph -> ta )	theorem	set	[]	set.mm	4syl	Inference chaining three syllogisms ~ syl . (Contributed by BJ, 14-Jul-2018.) The use of this theorem is marked "discouraged" because it can cause the Metamath program "MM-PA> MINIMIZE__WITH *" command to have very long run times. However, feel free to use "MM-PA> MINIMIZE__WITH 4syl / OVERRIDE" if you wish. Remember to update the "discouraged" file if it gets used. (New usage is discouraged.)
mpi : \|- ( ph -> ch )	theorem	set	[]	set.mm	mpi	null
mpisyl : \|- ( ph -> th )	theorem	set	[]	set.mm	mpisyl	A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.)
id : \|- ( ph -> ph )	theorem	set	[]	set.mm	id	null
idALT : \|- ( ph -> ph )	theorem	set	[]	set.mm	idALT	null
idd : \|- ( ph -> ( ps -> ps ) )	theorem	set	[]	set.mm	idd	Principle of identity ~ id with antecedent. (Contributed by NM, 26-Nov-1995.)
a1d : \|- ( ph -> ( ch -> ps ) )	theorem	set	[]	set.mm	a1d	null
2a1d : \|- ( ph -> ( ch -> ( th -> ps ) ) )	theorem	set	[]	set.mm	2a1d	Deduction introducing two antecedents. Two applications of ~ a1d . Deduction associated with ~ 2a1 and ~ 2a1i . (Contributed by BJ, 10-Aug-2020.)
a1i13 : \|- ( ph -> ( ps -> ( ch -> th ) ) )	theorem	set	[]	set.mm	a1i13	Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
2a1 : \|- ( ph -> ( ps -> ( ch -> ph ) ) )	theorem	set	[]	set.mm	2a1	A double form of ~ ax-1 . Its associated inference is ~ 2a1i . Its associated deduction is ~ 2a1d . (Contributed by BJ, 10-Aug-2020.) (Proof shortened by Wolf Lammen, 1-Sep-2020.)
a2d : \|- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) )	theorem	set	[]	set.mm	a2d	Deduction distributing an embedded antecedent. Deduction form of ~ ax-2 . (Contributed by NM, 23-Jun-1994.)
sylcom : \|- ( ph -> ( ps -> th ) )	theorem	set	[]	set.mm	sylcom	Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)
syl5com : \|- ( ph -> ( ch -> th ) )	theorem	set	[]	set.mm	syl5com	Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.)
com12 : \|- ( ps -> ( ph -> ch ) )	theorem	set	[]	set.mm	com12	null
syl11 : \|- ( ps -> ( th -> ch ) )	theorem	set	[]	set.mm	syl11	A syllogism inference. Commuted form of an instance of ~ syl . (Contributed by BJ, 25-Oct-2021.)
syl5 : \|- ( ch -> ( ph -> th ) )	theorem	set	[]	set.mm	syl5	A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.)
syl6 : \|- ( ph -> ( ps -> th ) )	theorem	set	[]	set.mm	syl6	A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Wolf Lammen, 30-Jul-2012.)
syl56 : \|- ( ch -> ( ph -> ta ) )	theorem	set	[]	set.mm	syl56	Combine ~ syl5 and ~ syl6 . (Contributed by NM, 14-Nov-2013.)
syl6com : \|- ( ps -> ( ph -> th ) )	theorem	set	[]	set.mm	syl6com	Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
mpcom : \|- ( ps -> ch )	theorem	set	[]	set.mm	mpcom	Modus ponens inference with commutation of antecedents. Commuted form of ~ mpd . (Contributed by NM, 17-Mar-1996.)
syli : \|- ( ps -> ( ph -> th ) )	theorem	set	[]	set.mm	syli	Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.)
syl2im : \|- ( ph -> ( ch -> ta ) )	theorem	set	[]	set.mm	syl2im	Replace two antecedents. Implication-only version of ~ syl2an . (Contributed by Wolf Lammen, 14-May-2013.)
syl2imc : \|- ( ch -> ( ph -> ta ) )	theorem	set	[]	set.mm	syl2imc	A commuted version of ~ syl2im . Implication-only version of ~ syl2anr . (Contributed by BJ, 20-Oct-2021.)
pm2.27 : \|- ( ph -> ( ( ph -> ps ) -> ps ) )	theorem	set	[]	set.mm	pm2.27	null
mpdd : \|- ( ph -> ( ps -> th ) )	theorem	set	[]	set.mm	mpdd	A nested modus ponens deduction. Double deduction associated with ~ ax-mp . Deduction associated with ~ mpd . (Contributed by NM, 12-Dec-2004.)
mpid : \|- ( ph -> ( ps -> th ) )	theorem	set	[]	set.mm	mpid	A nested modus ponens deduction. Deduction associated with ~ mpi . (Contributed by NM, 14-Dec-2004.)
mpdi : \|- ( ph -> ( ps -> th ) )	theorem	set	[]	set.mm	mpdi	A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.)
mpii : \|- ( ph -> ( ps -> th ) )	theorem	set	[]	set.mm	mpii	A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.)
syld : \|- ( ph -> ( ps -> th ) )	theorem	set	[]	set.mm	syld	Syllogism deduction. Deduction associated with ~ syl . See ~ conventions for the meaning of "associated deduction" or "deduction form". (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
syldc : \|- ( ps -> ( ph -> th ) )	theorem	set	[]	set.mm	syldc	Syllogism deduction. Commuted form of ~ syld . (Contributed by BJ, 25-Oct-2021.)
mp2d : \|- ( ph -> th )	theorem	set	[]	set.mm	mp2d	null
a1dd : \|- ( ph -> ( ps -> ( th -> ch ) ) )	theorem	set	[]	set.mm	a1dd	Double deduction introducing an antecedent. Deduction associated with ~ a1d . Double deduction associated with ~ ax-1 and ~ a1i . (Contributed by NM, 17-Dec-2004.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.)
2a1dd : \|- ( ph -> ( ps -> ( th -> ( ta -> ch ) ) ) )	theorem	set	[]	set.mm	2a1dd	Double deduction introducing two antecedents. Two applications of ~ 2a1dd . Deduction associated with ~ 2a1d . Double deduction associated with ~ 2a1 and ~ 2a1i . (Contributed by Jeff Hankins, 5-Aug-2009.)
pm2.43i : \|- ( ph -> ps )	theorem	set	[]	set.mm	pm2.43i	Inference absorbing redundant antecedent. Inference associated with ~ pm2.43 . (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)
pm2.43d : \|- ( ph -> ( ps -> ch ) )	theorem	set	[]	set.mm	pm2.43d	Deduction absorbing redundant antecedent. Deduction associated with ~ pm2.43 and ~ pm2.43i . (Contributed by NM, 18-Aug-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)
pm2.43a : \|- ( ps -> ( ph -> ch ) )	theorem	set	[]	set.mm	pm2.43a	Inference absorbing redundant antecedent. (Contributed by NM, 7-Nov-1995.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)
pm2.43b : \|- ( ph -> ( ps -> ch ) )	theorem	set	[]	set.mm	pm2.43b	Inference absorbing redundant antecedent. (Contributed by NM, 31-Oct-1995.)
pm2.43 : \|- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )	theorem	set	[]	set.mm	pm2.43	null
imim2d : \|- ( ph -> ( ( th -> ps ) -> ( th -> ch ) ) )	theorem	set	[]	set.mm	imim2d	Deduction adding nested antecedents. Deduction associated with ~ imim2 and ~ imim2i . (Contributed by NM, 10-Jan-1993.)
imim2 : \|- ( ( ph -> ps ) -> ( ( ch -> ph ) -> ( ch -> ps ) ) )	theorem	set	[]	set.mm	imim2	null
embantd : \|- ( ph -> ( ( ps -> ch ) -> th ) )	theorem	set	[]	set.mm	embantd	Deduction embedding an antecedent. (Contributed by Wolf Lammen, 4-Oct-2013.)
3syld : \|- ( ph -> ( ps -> ta ) )	theorem	set	[]	set.mm	3syld	Triple syllogism deduction. Deduction associated with ~ 3syld . (Contributed by Jeff Hankins, 4-Aug-2009.)
sylsyld : \|- ( ph -> ( ch -> ta ) )	theorem	set	[]	set.mm	sylsyld	A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)
imim12i : \|- ( ( ps -> ch ) -> ( ph -> th ) )	theorem	set	[]	set.mm	imim12i	Inference joining two implications. Inference associated with ~ imim12 . Its associated inference is ~ 3syl . (Contributed by NM, 12-Mar-1993.) (Proof shortened by Mel L. O'Cat, 29-Oct-2011.)
imim1i : \|- ( ( ps -> ch ) -> ( ph -> ch ) )	theorem	set	[]	set.mm	imim1i	null
imim3i : \|- ( ( th -> ph ) -> ( ( th -> ps ) -> ( th -> ch ) ) )	theorem	set	[]	set.mm	imim3i	Inference adding three nested antecedents. (Contributed by NM, 19-Dec-2006.)
sylc : \|- ( ph -> th )	theorem	set	[]	set.mm	sylc	A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994.) (Revised by NM, 13-Jul-2013.)
syl3c : \|- ( ph -> ta )	theorem	set	[]	set.mm	syl3c	A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.)
syl6mpi : \|- ( ph -> ( ps -> ta ) )	theorem	set	[]	set.mm	syl6mpi	A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)
mpsyl : \|- ( ps -> th )	theorem	set	[]	set.mm	mpsyl	Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)
mpsylsyld : \|- ( ps -> ( ch -> ta ) )	theorem	set	[]	set.mm	mpsylsyld	Modus ponens combined with a double syllogism inference. (Contributed by Alan Sare, 22-Jul-2012.)
syl6c : \|- ( ph -> ( ps -> ta ) )	theorem	set	[]	set.mm	syl6c	Inference combining ~ syl6 with contraction. (Contributed by Alan Sare, 2-May-2011.)
syl6ci : \|- ( ph -> ( ps -> ta ) )	theorem	set	[]	set.mm	syl6ci	A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)
syldd : \|- ( ph -> ( ps -> ( ch -> ta ) ) )	theorem	set	[]	set.mm	syldd	Nested syllogism deduction. Deduction associated with ~ syld . Double deduction associated with ~ syl . (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.)
syl5d : \|- ( ph -> ( th -> ( ps -> ta ) ) )	theorem	set	[]	set.mm	syl5d	A nested syllogism deduction. Deduction associated with ~ syl5 . (Contributed by NM, 14-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.)
syl7 : \|- ( ch -> ( th -> ( ph -> ta ) ) )	theorem	set	[]	set.mm	syl7	A syllogism rule of inference. The first premise is used to replace the third antecedent of the second premise. (Contributed by NM, 12-Jan-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
syl6d : \|- ( ph -> ( ps -> ( ch -> ta ) ) )	theorem	set	[]	set.mm	syl6d	A nested syllogism deduction. Deduction associated with ~ syl6 . (Contributed by NM, 11-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.)
syl8 : \|- ( ph -> ( ps -> ( ch -> ta ) ) )	theorem	set	[]	set.mm	syl8	A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
syl9 : \|- ( ph -> ( th -> ( ps -> ta ) ) )	theorem	set	[]	set.mm	syl9	A nested syllogism inference with different antecedents. (Contributed by NM, 13-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
syl9r : \|- ( th -> ( ph -> ( ps -> ta ) ) )	theorem	set	[]	set.mm	syl9r	A nested syllogism inference with different antecedents. (Contributed by NM, 14-May-1993.)
syl10 : \|- ( ph -> ( ps -> ( th -> et ) ) )	theorem	set	[]	set.mm	syl10	A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)
a1ddd : \|- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )	theorem	set	[]	set.mm	a1ddd	Triple deduction introducing an antecedent to a wff. Deduction associated with ~ a1dd . Double deduction associated with ~ a1d . Triple deduction associated with ~ ax-1 and ~ a1i . (Contributed by Jeff Hankins, 4-Aug-2009.)
imim12d : \|- ( ph -> ( ( ch -> th ) -> ( ps -> ta ) ) )	theorem	set	[]	set.mm	imim12d	Deduction combining antecedents and consequents. Deduction associated with ~ imim12 and ~ imim12i . (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mel L. O'Cat, 30-Oct-2011.)
imim1d : \|- ( ph -> ( ( ch -> th ) -> ( ps -> th ) ) )	theorem	set	[]	set.mm	imim1d	Deduction adding nested consequents. Deduction associated with ~ imim1 and ~ imim1i . (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)
imim1 : \|- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )	theorem	set	[]	set.mm	imim1	null
pm2.83 : \|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ( ch -> th ) ) -> ( ph -> ( ps -> th ) ) ) )	theorem	set	[]	set.mm	pm2.83	Theorem *2.83 of [WhiteheadRussell] p. 108. Closed form of ~ syld . (Contributed by NM, 3-Jan-2005.)
peirceroll : \|- ( ( ( ( ph -> ps ) -> ph ) -> ph ) -> ( ( ( ph -> ps ) -> ch ) -> ( ( ch -> ph ) -> ph ) ) )	theorem	set	[]	set.mm	peirceroll	Over minimal implicational calculus, Peirce's axiom ~ peirce implies an axiom sometimes called "Roll", ` ( ( ( ph -> ps ) -> ch ) -> ( ( ch -> ph ) -> ph ) ) ` , of which ~ looinv is a special instance. The converse also holds: substitute ` ( ph -> ps ) ` for ` ch ` in Roll and use ~ id and ~ ax-mp . (Contributed by BJ, 15-Jun-2021.)
com23 : \|- ( ph -> ( ch -> ( ps -> th ) ) )	theorem	set	[]	set.mm	com23	Commutation of antecedents. Swap 2nd and 3rd. Deduction associated with ~ com12 . (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
com3r : \|- ( ch -> ( ph -> ( ps -> th ) ) )	theorem	set	[]	set.mm	com3r	Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)
com13 : \|- ( ch -> ( ps -> ( ph -> th ) ) )	theorem	set	[]	set.mm	com13	Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
com3l : \|- ( ps -> ( ch -> ( ph -> th ) ) )	theorem	set	[]	set.mm	com3l	Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
pm2.04 : \|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )	theorem	set	[]	set.mm	pm2.04	null
com34 : \|- ( ph -> ( ps -> ( th -> ( ch -> ta ) ) ) )	theorem	set	[]	set.mm	com34	Commutation of antecedents. Swap 3rd and 4th. Deduction associated with ~ com23 . Double deduction associated with ~ com12 . (Contributed by NM, 25-Apr-1994.)
com4l : \|- ( ps -> ( ch -> ( th -> ( ph -> ta ) ) ) )	theorem	set	[]	set.mm	com4l	Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Mel L. O'Cat, 15-Aug-2004.)
com4t : \|- ( ch -> ( th -> ( ph -> ( ps -> ta ) ) ) )	theorem	set	[]	set.mm	com4t	Commutation of antecedents. Rotate twice. (Contributed by NM, 25-Apr-1994.)
com4r : \|- ( th -> ( ph -> ( ps -> ( ch -> ta ) ) ) )	theorem	set	[]	set.mm	com4r	Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)
com24 : \|- ( ph -> ( th -> ( ch -> ( ps -> ta ) ) ) )	theorem	set	[]	set.mm	com24	Commutation of antecedents. Swap 2nd and 4th. Deduction associated with ~ com13 . (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
com14 : \|- ( th -> ( ps -> ( ch -> ( ph -> ta ) ) ) )	theorem	set	[]	set.mm	com14	Commutation of antecedents. Swap 1st and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
com45 : \|- ( ph -> ( ps -> ( ch -> ( ta -> ( th -> et ) ) ) ) )	theorem	set	[]	set.mm	com45	Commutation of antecedents. Swap 4th and 5th. Deduction associated with ~ com34 . Double deduction associated with ~ com23 . Triple deduction associated with ~ com12 . (Contributed by Jeff Hankins, 28-Jun-2009.)
com35 : \|- ( ph -> ( ps -> ( ta -> ( th -> ( ch -> et ) ) ) ) )	theorem	set	[]	set.mm	com35	Commutation of antecedents. Swap 3rd and 5th. Deduction associated with ~ com24 . Double deduction associated with ~ com13 . (Contributed by Jeff Hankins, 28-Jun-2009.)
com25 : \|- ( ph -> ( ta -> ( ch -> ( th -> ( ps -> et ) ) ) ) )	theorem	set	[]	set.mm	com25	Commutation of antecedents. Swap 2nd and 5th. Deduction associated with ~ com14 . (Contributed by Jeff Hankins, 28-Jun-2009.)
com5l : \|- ( ps -> ( ch -> ( th -> ( ta -> ( ph -> et ) ) ) ) )	theorem	set	[]	set.mm	com5l	Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)
com15 : \|- ( ta -> ( ps -> ( ch -> ( th -> ( ph -> et ) ) ) ) )	theorem	set	[]	set.mm	com15	Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)
com52l : \|- ( ch -> ( th -> ( ta -> ( ph -> ( ps -> et ) ) ) ) )	theorem	set	[]	set.mm	com52l	Commutation of antecedents. Rotate left twice. (Contributed by Jeff Hankins, 28-Jun-2009.)
com52r : \|- ( th -> ( ta -> ( ph -> ( ps -> ( ch -> et ) ) ) ) )	theorem	set	[]	set.mm	com52r	Commutation of antecedents. Rotate right twice. (Contributed by Jeff Hankins, 28-Jun-2009.)
com5r : \|- ( ta -> ( ph -> ( ps -> ( ch -> ( th -> et ) ) ) ) )	theorem	set	[]	set.mm	com5r	Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen, 29-Jul-2012.)
imim12 : \|- ( ( ph -> ps ) -> ( ( ch -> th ) -> ( ( ps -> ch ) -> ( ph -> th ) ) ) )	theorem	set	[]	set.mm	imim12	Closed form of ~ imim12i and of ~ 3syl . (Contributed by BJ, 16-Jul-2019.)
jarr : \|- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) )	theorem	set	[]	set.mm	jarr	Elimination of a nested antecedent. Sometimes called "Syll-Simp" since it is a syllogism applied to ~ ax-1 ("Simplification"). (Contributed by Wolf Lammen, 9-May-2013.)

idi : |- ph

theorem

set

[]

set.mm

idi

This is the Metamath database set.mm. $) $( Metamath is a formal language and associated computer program for archiving, verifying, and studying mathematical proofs, created by Norman Dwight Megill (1950--2021). For more information, visit https://us.metamath.org and https://github.com/metamath/set.mm, and feel free to ask questions at https://groups.google.com/g/metamath. $) $( New users may want to read https://us.metamath.org/mpeuni/conventions.html to understand the label naming conventions used in set.mm. See also the Metamath program command "MM> HELP VERIFY MARKUP" for markup conventions. $) $( To break this file into smaller modules, in the Metamath program type "MM> READ set.mm" followed by "MM> WRITE SOURCE set.mm / SPLIT". To recombine, omit "/ SPLIT". $) $( The database set.mm was created by Norman Megill on 30-Sep-1992 and has been continuously enriched since then (list of contributors below). $) $( ! #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# Metamath source file for logic and set theory #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# ~~ PUBLIC DOMAIN ~~ This work is waived of all rights, including copyright, according to the CC0 Public Domain Dedication. https://creativecommons.org/publicdomain/zero/1.0/ Currently active maintainers: See the list in the CONTRIBUTING.md file of https://github.com/metamath/set.mm. Contributor list: DA David Abernethy SA Stefan Allan TA Thierry Arnoux JA Juha Arpiainen JB Jonathan Ben-Naim GB Gregory Bush MC Mario Carneiro FC Filip Cernatescu PC Paul Chapman DF Drahflow AD Adrian Ducourtial GD Georgy Dunaev SF Scott Fenton GG Gino Giotto JGH Jeff Hankins AH Anthony Hart DH David Harvey CH Chen-Pang He JH Jeff Hoffman II Igor Ieskov AI Asger C. Ipsen JJ Jerry James SJ Szymon Jaroszewicz BJ Benoit Jubin JK Jim Kingdon ML M L WL Wolf Lammen GL Gerard Lang BL Brendan Leahy LL Larry Lesyna RL Raph Levien FL Frederic Line RFL Roy F. Longton TM T M JPM Jeffrey P. Machado JM Jeff Madsen GM Giovanni Mascellani PM Peter Mazsa RM Rodolfo Medina NM Norman Megill MKU metakunt DM David Moews MM Mykola Mostovenko SN Steven Nguyen MO Mel L. O'Cat OAI OpenAI SO Stefan O'Rear JO Jason Orendorff KP K P NP Noam Pasman JPP Jon Pennant RP Richard Penner SP Stanislas Polu JP Josh Purinton RMI Remi RR Rohan Ridenour SR Steve Rodriguez ATS Andrew Salmon AS Alan Sare ES Eric Schmidt GS Glauco Siliprandi SS Saveliy Skresanov BT BTernaryTau ET Ender Ting JU Jarvin Udandy ADH Stijn "Adhemar" Vandamme AV Alexander van der Vekens JV Jannik Vierling ZW Zhi Wang EW Emmett Weisz DAW David A. Wheeler RW Roger Witte KW Kyle Wyonch JY Jonathan Yan FZ Fan Zheng KZ Kunhao Zheng HTML code for accented names: BJ Benoît Jubin GL Gérard Lang FL Frédéric Liné $) $( See "MM> HELP VERIFY MARKUP" for help with modularization tags. $) $( Begin $[ set-header.mm $] $) $( ! =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Contents of this header =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= * Quick "How To" * Bibliography * Metamath syntax summary * Other notes =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Quick "How To" =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= How to use this file under Windows 95/98/NT/2K/XP/Vista/7/10: 1. Download the Metamath program metamath.exe following the instructions on the Metamath home page (https://us.metamath.org) and put it in the same directory as this file. 2. In Windows Explorer, double-click on metamath.exe. 3. Type "read set.mm" and press Enter. 4. Type "help" for a list of help topics, and "help demo" for some command examples. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Bibliography =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Bibliographical references are made by bracketing an identifier in a theorem's comment, such as [RussellWhitehead]. These refer to HTML tags on the following web pages: Logic and set theory - see https://us.metamath.org/mpeuni/mmset.html#bib Hilbert space - see https://us.metamath.org/mpeuni/mmhil.html#ref A bracketed reference must be preceded by a theorem number, etc. and followed by a page number. See "MM> HELP WRITE BIBLIOGRAPHY" for details. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Metamath syntax summary =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The HELP LANGUAGE command in the Metamath program will give you a quick overview of Metamath. The specification is found on pp. 111--114 of the Metamath book. The following syntax summary is provided for convenience but may omit some details. A Metamath database (set of one or more ASCII source files) is a sequence of _tokens_, which are normally separated by spaces or line breaks. The only tokens that are built into the Metamath language are those (two-character sequences) beginning with $, shown in the following. These tokens are called _keywords_: $c ... $. - Constant declaration $v ... $. - Variable declaration $d ... $. - Disjoint (distinct) variable restriction <label> $f ... $. - "Floating" hypothesis (i.e. variable type declaration) <label> $e ... $. - "Essential" hypothesis (i.e. a logical assumption for a theorem or axiom) <label> $a ... $. - Axiom or definition or syntax construction <label> $p ... $= ... $. - Theorem and its proof ${ ... $} - Block for defining the scope of the above statements (except $a, $p which are forever active) $) $( ... $) $( - Comments (may not be nested); see HELP LANGUAGE for markup features. $[ ... $] - Include a file The above two-character sequences beginning with "$" are the only primitives built into the Metamath language. The only "logic" Metamath uses in its proof verification algorithm is the substitution of expressions for variables while checking for distinct variable violations. Everything else, including the axioms for logic, is defined in this database file. All other tokens are user-defined, and their names are arbitrary. There are two kinds of user-defined tokens, called math symbols (or just symbols) and labels. A _symbol_ may contain any non-whitespace printable character except "$". A _label_ may contain only alphanumeric characters and the characters "." (period), "-" (hyphen), and "_" (underscore). Symbols and labels are case-sensitive. All labels (except in proofs) must be distinct. A label may not have the same name as a symbol (to simplify the coding of certain parsers and translators). Here is some more detail about the syntax: $c <symbollist> $. <symbollist> is a (whitespace-separated) list of distinct symbols that haven't been used before. $v <symbollist> $. <symbollist> is a list of distinct symbols that haven't been used yet in the current scope (see ${ ... $} below). $d <symbollist> $. <symbollist> is a (whitespace-separated) list of distinct symbols previously declared with $v in current scope. It means that substitutions into these symbols may not have variables in common. <label> $f <symbollist> $. <symbollist> is a list of 2 symbols, the first of which must be previously declared with $c in the current scope. <label> $e <symbollist> $. <symbollist> is a list of 2 or more symbols, the first of which must be previously declared with $c in the current scope. <label> $a <symbollist> $. <symbollist> is a list of 2 or more symbols, the first of which must be previously declared with $c in the current scope. <label> $p <symbollist> $= <proof> $. <symbollist> is a list of 2 or more symbols, the first of which must be previously declared with $c in the current scope. <proof> is either a whitespace-delimited sequence of previous labels (created by SAVE PROOF <label> /NORMAL) or a compressed proof (created by SAVE PROOF <label> /COMPRESSED). After using SAVE PROOF, use WRITE SOURCE to save the database file to disk. ${ ... $} Block for scoping the above statements (except $a, $p which are forever active). Currently, $c may not occur inside of a block. $) $( <any text> $) $( Comment. Note: <any text> may not contain adjacent "$" and ")" characters. The comment opening and closing delimiters must be surrounded by whitespace (space, tab, CR, LF, or FF). $[ <filename> $] Insert contents of <filename> at this point. If <filename> is current file or has been already been inserted, it will not be inserted again. Inside of comments, it is recommended that labels be preceded with a tilde (~) and math symbol tokens be enclosed in grave accents, also known as backticks (` `). These tildes, tokens, math symbols and backticks should be surrounded by spaces. This way the LaTeX and HTML rendition of comments will be accurate, and tools to globally change labels and math symbols will also change them in comments. Note that inside of backticks a pair of backticks is interpreted as a single backtick. A special comment containing $ t (with no space after the dollar sign) defines LaTeX and HTML symbols. See HELP LANGUAGE and HELP HTML for other markup features in comments. The proofs in this file are in "compressed" format for storage efficiency. The Metamath program reads the compressed format directly. This format is described in Appendix B of the Metamath book. It is not intended to be read by humans. For viewing proofs you should use the various SHOW PROOF commands described in the Metamath book (or the online HELP). The Metamath program does not normally affect any content of this file other than proofs, i.e., the text between "$=" and "$." (and some rewrapping). All other content is user-created. Proofs are created or modified with the PROVE command. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Other notes =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 1. It is recommended that you be familiar with Chapters 2 and 4 of the Metamath book to understand the Metamath language. Chapters 2, 3 and 5 explain how to use the Metamath program. Chapter 3 gives an informal overview of what this source file is all about. Appendix A gives the standard mathematical symbols corresponding to some of the ASCII tokens used in this file. The ASCII tokens may seem cryptic at first, even if you are familiar with set theory, but a review of the definition summary in Chapter 3 should quickly enable you to see the correspondence to standard mathematical notation. To easily find the definition of a token, search for the first occurrences of the token surrounded by spaces. Some odd-looking ones include "-." for "not", and "C_" for "is a subset of". The Metamath program "MM> HELP TEX" command explains how to obtain a LaTeX output to see the real mathematical symbols. Let us know if you have better suggestions for naming ASCII tokens. 2. Theorems can be written in different forms, including "closed form", "deduction form", and "inference form" (for details, see ~ conventions ). For basic theorems, all three forms are generally given, but for more advanced theorems, we prefer to use the deduction form, since it permits to write proofs in the "deduction style", and we do not add theorems in inference form unless there are reasonable grounds for it (for instance, shortening sufficiently many proofs to counterbalance their addition). 3. On providing new definitions and theorems, the conventions provided in the comment of ~ conventions should be obeyed. 4. For a chronological list of changes to label names and label deletions, see the changes-set.txt file. This should help if you have a proof not checked into the main repository and want to update it for recent changes. $) $( End $[ set-header.mm $] $) $( Begin $[ set-main.mm $] $) $( Begin $[ set-pred.mm $] $) $( The following header is the first to appear in the Theorem List contents, because higher-level headers suppress all previous same-level or lower-level headers in the same comment area between $a and $p statements. See "MM> HELP WRITE THEOREM_LIST" for information about headers. $) $( ############################################################################### CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY ############################################################################### Logic can be defined as the "study of the principles of correct reasoning" (Merrilee H. Salmon's 1991 "Informal Reasoning and Informal Logic" in _Informal Reasoning and Education_) or as "a formal system using symbolic techniques and mathematical methods to establish truth-values" (the Oxford English Dictionary). This section formally defines the logic system we will use. In particular, it defines symbols for declaring truthful statements, along with rules for deriving truthful statements from other truthful statements. The system defined here is classical first-order logic (often abbreviated as FOL) with equality and no terms (the most common logic system used by mathematicians). We begin with a few housekeeping items in pre-logic, and then introduce propositional calculus (both its axioms and important theorems that can be derived from them). Propositional calculus deals with general truths about well-formed formulas (wffs) regardless of how they are constructed. This is followed by proofs that other axiomatizations of classical propositional calculus can be derived from the axioms we have chosen to use. We then define predicate calculus, which adds additional symbols and rules useful for discussing objects (beyond simply true or false). In particular, it introduces the symbols ` = ` ("equals"), ` e. ` ("is a member of"), and ` A. ` ("for all"). The first two are called "predicates". A predicate specifies a true or false relationship between its two arguments. $) $( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# Pre-logic #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# This section includes a few "housekeeping" mechanisms before we begin defining the basics of logic. $) $( Declare the primitive constant symbols for propositional calculus. $) $c ( $. $( Left parenthesis $) $c ) $. $( Right parenthesis $) $c -> $. $( Right arrow (read: "implies") $) $c -. $. $( Right handle (read: "not") $) $c wff $. $( Well-formed formula symbol (read: "the following symbol sequence is a wff") $) $c |- $. $( Turnstile (read: "the following symbol sequence is provable" or "a proof exists for") $) $( Define the syntax and logical typecodes, and declare that our grammar is unambiguous (verifiable using the KLR parser, with compositing depth 5). (This $ j comment need not be read by verifiers, but is useful for parsers like mmj2.) $) $( $j syntax 'wff'; syntax '|-' as 'wff'; unambiguous 'klr 5'; $) $( Declare the color of wff variables. $) $( $j varcolorcode "wff" as "0000FF"; altvarcolorcode "wff" as "337DFF"; $) $( Declare typographical constant symbols that are not directly used in the formalism but are useful to explain it in comments. $) $c & $. $( Ampersand (read: "and"). $) $c => $. $( Double right arrow (read: "implies"). $) $( wff variable sequence: ph ps ch th ta et ze si rh mu la ka $) $( Introduce some variable names we will use to represent well-formed formulas (wff's). $) $v ph $. $( Greek phi $) $v ps $. $( Greek psi $) $v ch $. $( Greek chi $) $v th $. $( Greek theta $) $v ta $. $( Greek tau $) $v et $. $( Greek eta $) $v ze $. $( Greek zeta $) $v si $. $( Greek sigma $) $v rh $. $( Greek rho $) $v mu $. $( Greek mu $) $v la $. $( Greek lambda $) $v ka $. $( Greek kappa $) $( Specify some variables that we will use to represent wff's. The fact that a variable represents a wff is relevant only to a theorem referring to that variable, so we may use $f hypotheses. The symbol ` wff ` specifies that the variable that follows it represents a wff. $) $( Let variable ` ph ` be a wff. $) wph $f wff ph $. $( Let variable ` ps ` be a wff. $) wps $f wff ps $. $( Let variable ` ch ` be a wff. $) wch $f wff ch $. $( Let variable ` th ` be a wff. $) wth $f wff th $. $( Let variable ` ta ` be a wff. $) wta $f wff ta $. $( Let variable ` et ` be a wff. $) wet $f wff et $. $( Let variable ` ze ` be a wff. $) wze $f wff ze $. $( Let variable ` si ` be a wff. $) wsi $f wff si $. $( Let variable ` rh ` be a wff. $) wrh $f wff rh $. $( Let variable ` mu ` be a wff. $) wmu $f wff mu $. $( Let variable ` la ` be a wff. $) wla $f wff la $. $( Let variable ` ka ` be a wff. $) wka $f wff ka $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Inferences for assisting proof development =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The inference rules in this section will normally never appear in a completed proof. They can be ignored if you are using this database to assist learning logic - please start with the statement ~ wn instead. $) ${ idi.1 $e |- ph $. $( (_Note_: This inference rule and the next one, ~ a1ii , will normally never appear in a completed proof. They can be ignored if you are using this database to assist learning logic; please start with the statement ~ wn instead.) This inference says "if ` ph ` is true then ` ph ` is true". This inference requires no axioms for its proof, and is useful as a copy-paste mechanism during proof development in mmj2. It is normally not referenced in the final version of a proof, since it is always redundant. You can remove this using the metamath-exe (Metamath program) Proof Assistant using the "MM-PA> MINIMIZE__WITH *" command. This is the inference associated with ~ id , hence its name. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

a1ii : |- ph

theorem

set

[]

set.mm

a1ii

(_Note_: This inference rule and the previous one, ~ idi , will normally never appear in a completed proof.) This is a technical inference to assist proof development. It provides a temporary way to add an independent subproof to a proof under development, for later assignment to a normal proof step. The Metamath (Metamath-exe) program Proof Assistant requires proofs to be developed backwards from the conclusion with no gaps, and it has no mechanism that lets the user work on isolated subproofs. This inference provides a workaround for this limitation. It can be inserted at any point in a proof to allow an independent subproof to be developed on the side, for later use as part of the final proof. _Instructions_: <HTML><ol><li>Assign this inference to any unknown step in the proof. Typically, the last unknown step is the most convenient, since <code>MM-PA> ASSIGN LAST</code> can be used. This step will be replicated in hypothesis a1ii.1, from where the development of the main proof can continue.</li><li>Develop the independent subproof backwards from hypothesis a1ii.2. If desired, use a <code>MM-PA> LET STEP</code> command to pre-assign the conclusion of the independent subproof to a1ii.2.</li><li>After the independent subproof is complete, use <code>MM-PA> IMPROVE ALL</code> to assign it automatically to an unknown step in the main proof that matches it.</li><li>After the entire proof is complete, use <code>MM-PA> MINIMIZE_WITH *</code> to clean up (discard) all ~ a1ii references automatically.</ol></HTML> This can also be used to apply subproofs from other theorems. In step 2, simply assign the theorem to a1ii.2, and run <HTML><code>MM-PA> EXPAND <theorem></code></HTML> to "import" a subproof from another theorem. This inference was originally designed to assist importing partially completed Proof Worksheets from the mmj2 Proof Assistant GUI, but it can also be useful on its own. Interestingly, no axioms are required for its proof. It is the inference associated with ~ a1i . (Contributed by NM, 7-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

mp2 : |- ch

theorem

set

[]

set.mm

mp2

Register implication '->' as a primitive expression (lacking a definition). $) $( $j primitive 'wi'; $) $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The axioms of propositional calculus =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Propositional calculus (Axioms ~ ax-1 through ~ ax-3 and rule ~ ax-mp ) can be thought of as asserting formulas that are universally "true" when their variables are replaced by any combination of "true" and "false". Propositional calculus was first formalized by Frege in 1879, using as his axioms (in addition to rule ~ ax-mp ) the wffs ~ ax-1 , ~ ax-2 , ~ pm2.04 , ~ con3 , ~ notnot , and ~ notnotr . Around 1930, Lukasiewicz simplified the system by eliminating the third (which follows from the first two, as you can see by looking at the proof of ~ pm2.04 ) and replacing the last three with our ~ ax-3 . (Thanks to Ted Ulrich for this information.) The theorems of propositional calculus are also called _tautologies_. Tautologies can be proved very simply using truth tables, based on the true/false interpretation of propositional calculus. To do this, we assign all possible combinations of true and false to the wff variables and verify that the result (using the rules described in ~ wi and ~ wn ) always evaluates to true. This is called the _semantic_ approach. Our approach is called the _syntactic_ approach, in which everything is derived from axioms. A metatheorem called the Completeness Theorem for Propositional Calculus shows that the two approaches are equivalent and even provides an algorithm for automatically generating syntactic proofs from a truth table. Those proofs, however, tend to be long, since truth tables grow exponentially with the number of variables, and the much shorter proofs that we show here were found manually. $) ${ $( Minor premise for modus ponens. $) min $e |- ph $. $( Major premise for modus ponens. $) maj $e |- ( ph -> ps ) $. $( Rule of Modus Ponens. The postulated inference rule of propositional calculus. See, e.g., Rule 1 of [Hamilton] p. 73. The rule says, "if ` ph ` is true, and ` ph ` implies ` ps ` , then ` ps ` must also be true". This rule is sometimes called "detachment", since it detaches the minor premise from the major premise. "Modus ponens" is short for "modus ponendo ponens", a Latin phrase that means "the mode that by affirming affirms" - remark in [Sanford] p. 39. This rule is similar to the rule of modus tollens ~ mto . Note: In some web page displays such as the Statement List, the symbols " ` & ` " and " ` => ` " informally indicate the relationship between the hypotheses and the assertion (conclusion), abbreviating the English words "and" and "implies". They are not part of the formal language. (Contributed by NM, 30-Sep-1992.) $) ax-mp $a |- ps $. $} $( Axiom _Simp_. Axiom A1 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. The 3 axioms are also given as Definition 2.1 of [Hamilton] p. 28. This axiom is called _Simp_ or "the principle of simplification" in _Principia Mathematica_ (Theorem *2.02 of [WhiteheadRussell] p. 100) because "it enables us to pass from the joint assertion of ` ph ` and ` ps ` to the assertion of ` ph ` simply". It is Proposition 1 of [Frege1879] p. 26, its first axiom. (Contributed by NM, 30-Sep-1992.) $) ax-1 $a |- ( ph -> ( ps -> ph ) ) $. $( Axiom _Frege_. Axiom A2 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It "distributes" an antecedent over two consequents. This axiom was part of Frege's original system and is known as _Frege_ in the literature; see Proposition 2 of [Frege1879] p. 26. It is also proved as Theorem *2.77 of [WhiteheadRussell] p. 108. The other direction of this axiom also turns out to be true, as demonstrated by ~ pm5.41 . (Contributed by NM, 30-Sep-1992.) $) ax-2 $a |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) $. $( Axiom _Transp_. Axiom A3 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky". This axiom is called _Transp_ or "the principle of transposition" in _Principia Mathematica_ (Theorem *2.17 of [WhiteheadRussell] p. 103). We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning. (Contributed by NM, 30-Sep-1992.) Use its alias ~ con4 instead. (New usage is discouraged.) $) ax-3 $a |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Logical implication =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The results in this section are based on implication only, and avoid ~ ax-3 , so are intuitionistic. The system { ~ ax-mp , ~ ax-1 , ~ ax-2 } axiomatizes what is sometimes called "intuitionistic implicational calculus" or "minimal implicational calculus". In an implication, the wff before the arrow is called the "antecedent" and the wff after the arrow is called the "consequent". $) ${ mp2.1 $e |- ph $. mp2.2 $e |- ps $. mp2.3 $e |- ( ph -> ( ps -> ch ) ) $. $( A double modus ponens inference. (Contributed by NM, 5-Apr-1994.)

mp2b : |- ch

theorem

set

[]

set.mm

mp2b

A double modus ponens inference. (Contributed by Mario Carneiro, 24-Jan-2013.)

a1i : |- ( ps -> ph )

theorem

set

[]

set.mm

a1i

null

2a1i : |- ( ps -> ( ch -> ph ) )

theorem

set

[]

set.mm

2a1i

Inference introducing two antecedents. Two applications of ~ a1i . Inference associated with ~ 2a1 . (Contributed by Jeff Hankins, 4-Aug-2009.)

mp1i : |- ( ch -> ps )

theorem

set

[]

set.mm

mp1i

Inference detaching an antecedent and introducing a new one. (Contributed by Stefan O'Rear, 29-Jan-2015.)

a2i : |- ( ( ph -> ps ) -> ( ph -> ch ) )

theorem

set

[]

set.mm

a2i

Inference distributing an antecedent. Inference associated with ~ ax-2 . Its associated inference is ~ mpd . (Contributed by NM, 29-Dec-1992.)

mpd : |- ( ph -> ch )

theorem

set

[]

set.mm

mpd

A modus ponens deduction. A translation of natural deduction rule ` -> ` E ( ` -> ` elimination), see ~ natded . Deduction form of ~ ax-mp . Inference associated with ~ a2i . Commuted form of ~ mpcom . (Contributed by NM, 29-Dec-1992.)

imim2i : |- ( ( ch -> ph ) -> ( ch -> ps ) )

theorem

set

[]

set.mm

imim2i

null

syl : |- ( ph -> ch )

theorem

set

[]

set.mm

syl

null

3syl : |- ( ph -> th )

theorem

set

[]

set.mm

3syl

Inference chaining two syllogisms ~ syl . Inference associated with ~ imim12i . (Contributed by NM, 28-Dec-1992.)

4syl : |- ( ph -> ta )

theorem

set

[]

set.mm

4syl

Inference chaining three syllogisms ~ syl . (Contributed by BJ, 14-Jul-2018.) The use of this theorem is marked "discouraged" because it can cause the Metamath program "MM-PA> MINIMIZE__WITH *" command to have very long run times. However, feel free to use "MM-PA> MINIMIZE__WITH 4syl / OVERRIDE" if you wish. Remember to update the "discouraged" file if it gets used. (New usage is discouraged.)

mpi : |- ( ph -> ch )

theorem

set

[]

set.mm

mpi

null

mpisyl : |- ( ph -> th )

theorem

set

[]

set.mm

mpisyl

A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.)

id : |- ( ph -> ph )

theorem

set

[]

set.mm

id

null

idALT : |- ( ph -> ph )

theorem

set

[]

set.mm

idALT

null

idd : |- ( ph -> ( ps -> ps ) )

theorem

set

[]

set.mm

idd

Principle of identity ~ id with antecedent. (Contributed by NM, 26-Nov-1995.)

a1d : |- ( ph -> ( ch -> ps ) )

theorem

set

[]

set.mm

a1d

null

2a1d : |- ( ph -> ( ch -> ( th -> ps ) ) )

theorem

set

[]

set.mm

2a1d

Deduction introducing two antecedents. Two applications of ~ a1d . Deduction associated with ~ 2a1 and ~ 2a1i . (Contributed by BJ, 10-Aug-2020.)

a1i13 : |- ( ph -> ( ps -> ( ch -> th ) ) )

theorem

set

[]

set.mm

a1i13

Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)

2a1 : |- ( ph -> ( ps -> ( ch -> ph ) ) )

theorem

set

[]

set.mm

2a1

A double form of ~ ax-1 . Its associated inference is ~ 2a1i . Its associated deduction is ~ 2a1d . (Contributed by BJ, 10-Aug-2020.) (Proof shortened by Wolf Lammen, 1-Sep-2020.)

a2d : |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) )

theorem

set

[]

set.mm

a2d

Deduction distributing an embedded antecedent. Deduction form of ~ ax-2 . (Contributed by NM, 23-Jun-1994.)

sylcom : |- ( ph -> ( ps -> th ) )

theorem

set

[]

set.mm

sylcom

Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)

syl5com : |- ( ph -> ( ch -> th ) )

theorem

set

[]

set.mm

syl5com

Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.)

com12 : |- ( ps -> ( ph -> ch ) )

theorem

set

[]

set.mm

com12

null

syl11 : |- ( ps -> ( th -> ch ) )

theorem

set

[]

set.mm

syl11

A syllogism inference. Commuted form of an instance of ~ syl . (Contributed by BJ, 25-Oct-2021.)

syl5 : |- ( ch -> ( ph -> th ) )

theorem

set

[]

set.mm

syl5

A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.)

syl6 : |- ( ph -> ( ps -> th ) )

theorem

set

[]

set.mm

syl6

A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Wolf Lammen, 30-Jul-2012.)

syl56 : |- ( ch -> ( ph -> ta ) )

theorem

set

[]

set.mm

syl56

Combine ~ syl5 and ~ syl6 . (Contributed by NM, 14-Nov-2013.)

syl6com : |- ( ps -> ( ph -> th ) )

theorem

set

[]

set.mm

syl6com

Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.)

mpcom : |- ( ps -> ch )

theorem

set

[]

set.mm

mpcom

Modus ponens inference with commutation of antecedents. Commuted form of ~ mpd . (Contributed by NM, 17-Mar-1996.)

syli : |- ( ps -> ( ph -> th ) )

theorem

set

[]

set.mm

syli

Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.)

syl2im : |- ( ph -> ( ch -> ta ) )

theorem

set

[]

set.mm

syl2im

Replace two antecedents. Implication-only version of ~ syl2an . (Contributed by Wolf Lammen, 14-May-2013.)

syl2imc : |- ( ch -> ( ph -> ta ) )

theorem

set

[]

set.mm

syl2imc

A commuted version of ~ syl2im . Implication-only version of ~ syl2anr . (Contributed by BJ, 20-Oct-2021.)

pm2.27 : |- ( ph -> ( ( ph -> ps ) -> ps ) )

theorem

set

[]

set.mm

pm2.27

null

mpdd : |- ( ph -> ( ps -> th ) )

theorem

set

[]

set.mm

mpdd

A nested modus ponens deduction. Double deduction associated with ~ ax-mp . Deduction associated with ~ mpd . (Contributed by NM, 12-Dec-2004.)

mpid : |- ( ph -> ( ps -> th ) )

theorem

set

[]

set.mm

mpid

A nested modus ponens deduction. Deduction associated with ~ mpi . (Contributed by NM, 14-Dec-2004.)

mpdi : |- ( ph -> ( ps -> th ) )

theorem

set

[]

set.mm

mpdi

A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.)

mpii : |- ( ph -> ( ps -> th ) )

theorem

set

[]

set.mm

mpii

A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.)

syld : |- ( ph -> ( ps -> th ) )

theorem

set

[]

set.mm

syld

Syllogism deduction. Deduction associated with ~ syl . See ~ conventions for the meaning of "associated deduction" or "deduction form". (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

syldc : |- ( ps -> ( ph -> th ) )

theorem

set

[]

set.mm

syldc

Syllogism deduction. Commuted form of ~ syld . (Contributed by BJ, 25-Oct-2021.)

mp2d : |- ( ph -> th )

theorem

set

[]

set.mm

mp2d

null

a1dd : |- ( ph -> ( ps -> ( th -> ch ) ) )

theorem

set

[]

set.mm

a1dd

Double deduction introducing an antecedent. Deduction associated with ~ a1d . Double deduction associated with ~ ax-1 and ~ a1i . (Contributed by NM, 17-Dec-2004.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.)

2a1dd : |- ( ph -> ( ps -> ( th -> ( ta -> ch ) ) ) )

theorem

set

[]

set.mm

2a1dd

Double deduction introducing two antecedents. Two applications of ~ 2a1dd . Deduction associated with ~ 2a1d . Double deduction associated with ~ 2a1 and ~ 2a1i . (Contributed by Jeff Hankins, 5-Aug-2009.)

pm2.43i : |- ( ph -> ps )

theorem

set

[]

set.mm

pm2.43i

Inference absorbing redundant antecedent. Inference associated with ~ pm2.43 . (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)

pm2.43d : |- ( ph -> ( ps -> ch ) )

theorem

set

[]

set.mm

pm2.43d

Deduction absorbing redundant antecedent. Deduction associated with ~ pm2.43 and ~ pm2.43i . (Contributed by NM, 18-Aug-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)

pm2.43a : |- ( ps -> ( ph -> ch ) )

theorem

set

[]

set.mm

pm2.43a

Inference absorbing redundant antecedent. (Contributed by NM, 7-Nov-1995.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)

pm2.43b : |- ( ph -> ( ps -> ch ) )

theorem

set

[]

set.mm

pm2.43b

Inference absorbing redundant antecedent. (Contributed by NM, 31-Oct-1995.)

pm2.43 : |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )

theorem

set

[]

set.mm

pm2.43

null

imim2d : |- ( ph -> ( ( th -> ps ) -> ( th -> ch ) ) )

theorem

set

[]

set.mm

imim2d

Deduction adding nested antecedents. Deduction associated with ~ imim2 and ~ imim2i . (Contributed by NM, 10-Jan-1993.)

imim2 : |- ( ( ph -> ps ) -> ( ( ch -> ph ) -> ( ch -> ps ) ) )

theorem

set

[]

set.mm

imim2

null

embantd : |- ( ph -> ( ( ps -> ch ) -> th ) )

theorem

set

[]

set.mm

embantd

Deduction embedding an antecedent. (Contributed by Wolf Lammen, 4-Oct-2013.)

3syld : |- ( ph -> ( ps -> ta ) )

theorem

set

[]

set.mm

3syld

Triple syllogism deduction. Deduction associated with ~ 3syld . (Contributed by Jeff Hankins, 4-Aug-2009.)

sylsyld : |- ( ph -> ( ch -> ta ) )

theorem

set

[]

set.mm

sylsyld

A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)

imim12i : |- ( ( ps -> ch ) -> ( ph -> th ) )

theorem

set

[]

set.mm

imim12i

Inference joining two implications. Inference associated with ~ imim12 . Its associated inference is ~ 3syl . (Contributed by NM, 12-Mar-1993.) (Proof shortened by Mel L. O'Cat, 29-Oct-2011.)

imim1i : |- ( ( ps -> ch ) -> ( ph -> ch ) )

theorem

set

[]

set.mm

imim1i

null

imim3i : |- ( ( th -> ph ) -> ( ( th -> ps ) -> ( th -> ch ) ) )

theorem

set

[]

set.mm

imim3i

Inference adding three nested antecedents. (Contributed by NM, 19-Dec-2006.)

sylc : |- ( ph -> th )

theorem

set

[]

set.mm

sylc

A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994.) (Revised by NM, 13-Jul-2013.)

syl3c : |- ( ph -> ta )

theorem

set

[]

set.mm

syl3c

A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.)

syl6mpi : |- ( ph -> ( ps -> ta ) )

theorem

set

[]

set.mm

syl6mpi

A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)

mpsyl : |- ( ps -> th )

theorem

set

[]

set.mm

mpsyl

Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)

mpsylsyld : |- ( ps -> ( ch -> ta ) )

theorem

set

[]

set.mm

mpsylsyld

Modus ponens combined with a double syllogism inference. (Contributed by Alan Sare, 22-Jul-2012.)

syl6c : |- ( ph -> ( ps -> ta ) )

theorem

set

[]

set.mm

syl6c

Inference combining ~ syl6 with contraction. (Contributed by Alan Sare, 2-May-2011.)

syl6ci : |- ( ph -> ( ps -> ta ) )

theorem

set

[]

set.mm

syl6ci

A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)

syldd : |- ( ph -> ( ps -> ( ch -> ta ) ) )

theorem

set

[]

set.mm

syldd

Nested syllogism deduction. Deduction associated with ~ syld . Double deduction associated with ~ syl . (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.)

syl5d : |- ( ph -> ( th -> ( ps -> ta ) ) )

theorem

set

[]

set.mm

syl5d

A nested syllogism deduction. Deduction associated with ~ syl5 . (Contributed by NM, 14-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.)

syl7 : |- ( ch -> ( th -> ( ph -> ta ) ) )

theorem

set

[]

set.mm

syl7

A syllogism rule of inference. The first premise is used to replace the third antecedent of the second premise. (Contributed by NM, 12-Jan-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

syl6d : |- ( ph -> ( ps -> ( ch -> ta ) ) )

theorem

set

[]

set.mm

syl6d

A nested syllogism deduction. Deduction associated with ~ syl6 . (Contributed by NM, 11-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.)

syl8 : |- ( ph -> ( ps -> ( ch -> ta ) ) )

theorem

set

[]

set.mm

syl8

A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

syl9 : |- ( ph -> ( th -> ( ps -> ta ) ) )

theorem

set

[]

set.mm

syl9

A nested syllogism inference with different antecedents. (Contributed by NM, 13-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)

syl9r : |- ( th -> ( ph -> ( ps -> ta ) ) )

theorem

set

[]

set.mm

syl9r

A nested syllogism inference with different antecedents. (Contributed by NM, 14-May-1993.)

syl10 : |- ( ph -> ( ps -> ( th -> et ) ) )

theorem

set

[]

set.mm

syl10

A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)

a1ddd : |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )

theorem

set

[]

set.mm

a1ddd

Triple deduction introducing an antecedent to a wff. Deduction associated with ~ a1dd . Double deduction associated with ~ a1d . Triple deduction associated with ~ ax-1 and ~ a1i . (Contributed by Jeff Hankins, 4-Aug-2009.)

imim12d : |- ( ph -> ( ( ch -> th ) -> ( ps -> ta ) ) )

theorem

set

[]

set.mm

imim12d

Deduction combining antecedents and consequents. Deduction associated with ~ imim12 and ~ imim12i . (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mel L. O'Cat, 30-Oct-2011.)

imim1d : |- ( ph -> ( ( ch -> th ) -> ( ps -> th ) ) )

theorem

set

[]

set.mm

imim1d

Deduction adding nested consequents. Deduction associated with ~ imim1 and ~ imim1i . (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)

imim1 : |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )

theorem

set

[]

set.mm

imim1

null

pm2.83 : |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ( ch -> th ) ) -> ( ph -> ( ps -> th ) ) ) )

theorem

set

[]

set.mm

pm2.83

Theorem *2.83 of [WhiteheadRussell] p. 108. Closed form of ~ syld . (Contributed by NM, 3-Jan-2005.)

peirceroll : |- ( ( ( ( ph -> ps ) -> ph ) -> ph ) -> ( ( ( ph -> ps ) -> ch ) -> ( ( ch -> ph ) -> ph ) ) )

theorem

set

[]

set.mm

peirceroll

Over minimal implicational calculus, Peirce's axiom ~ peirce implies an axiom sometimes called "Roll", ` ( ( ( ph -> ps ) -> ch ) -> ( ( ch -> ph ) -> ph ) ) ` , of which ~ looinv is a special instance. The converse also holds: substitute ` ( ph -> ps ) ` for ` ch ` in Roll and use ~ id and ~ ax-mp . (Contributed by BJ, 15-Jun-2021.)

com23 : |- ( ph -> ( ch -> ( ps -> th ) ) )

theorem

set

[]

set.mm

com23

Commutation of antecedents. Swap 2nd and 3rd. Deduction associated with ~ com12 . (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)

com3r : |- ( ch -> ( ph -> ( ps -> th ) ) )

theorem

set

[]

set.mm

com3r

Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)

com13 : |- ( ch -> ( ps -> ( ph -> th ) ) )

theorem

set

[]

set.mm

com13

Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

com3l : |- ( ps -> ( ch -> ( ph -> th ) ) )

theorem

set

[]

set.mm

com3l

Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

pm2.04 : |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )

theorem

set

[]

set.mm

pm2.04

null

com34 : |- ( ph -> ( ps -> ( th -> ( ch -> ta ) ) ) )

theorem

set

[]

set.mm

com34

Commutation of antecedents. Swap 3rd and 4th. Deduction associated with ~ com23 . Double deduction associated with ~ com12 . (Contributed by NM, 25-Apr-1994.)

com4l : |- ( ps -> ( ch -> ( th -> ( ph -> ta ) ) ) )

theorem

set

[]

set.mm

com4l

Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Mel L. O'Cat, 15-Aug-2004.)

com4t : |- ( ch -> ( th -> ( ph -> ( ps -> ta ) ) ) )

theorem

set

[]

set.mm

com4t

Commutation of antecedents. Rotate twice. (Contributed by NM, 25-Apr-1994.)

com4r : |- ( th -> ( ph -> ( ps -> ( ch -> ta ) ) ) )

theorem

set

[]

set.mm

com4r

Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)

com24 : |- ( ph -> ( th -> ( ch -> ( ps -> ta ) ) ) )

theorem

set

[]

set.mm

com24

Commutation of antecedents. Swap 2nd and 4th. Deduction associated with ~ com13 . (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

com14 : |- ( th -> ( ps -> ( ch -> ( ph -> ta ) ) ) )

theorem

set

[]

set.mm

com14

Commutation of antecedents. Swap 1st and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)

com45 : |- ( ph -> ( ps -> ( ch -> ( ta -> ( th -> et ) ) ) ) )

theorem

set

[]

set.mm

com45

Commutation of antecedents. Swap 4th and 5th. Deduction associated with ~ com34 . Double deduction associated with ~ com23 . Triple deduction associated with ~ com12 . (Contributed by Jeff Hankins, 28-Jun-2009.)

com35 : |- ( ph -> ( ps -> ( ta -> ( th -> ( ch -> et ) ) ) ) )

theorem

set

[]

set.mm

com35

Commutation of antecedents. Swap 3rd and 5th. Deduction associated with ~ com24 . Double deduction associated with ~ com13 . (Contributed by Jeff Hankins, 28-Jun-2009.)

com25 : |- ( ph -> ( ta -> ( ch -> ( th -> ( ps -> et ) ) ) ) )

theorem

set

[]

set.mm

com25

Commutation of antecedents. Swap 2nd and 5th. Deduction associated with ~ com14 . (Contributed by Jeff Hankins, 28-Jun-2009.)

com5l : |- ( ps -> ( ch -> ( th -> ( ta -> ( ph -> et ) ) ) ) )

theorem

set

[]

set.mm

com5l

Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)

com15 : |- ( ta -> ( ps -> ( ch -> ( th -> ( ph -> et ) ) ) ) )

theorem

set

[]

set.mm

com15

Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)

com52l : |- ( ch -> ( th -> ( ta -> ( ph -> ( ps -> et ) ) ) ) )

theorem

set

[]

set.mm

com52l

Commutation of antecedents. Rotate left twice. (Contributed by Jeff Hankins, 28-Jun-2009.)

com52r : |- ( th -> ( ta -> ( ph -> ( ps -> ( ch -> et ) ) ) ) )

theorem

set

[]

set.mm

com52r

Commutation of antecedents. Rotate right twice. (Contributed by Jeff Hankins, 28-Jun-2009.)

com5r : |- ( ta -> ( ph -> ( ps -> ( ch -> ( th -> et ) ) ) ) )

theorem

set

[]

set.mm

com5r

Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen, 29-Jul-2012.)

imim12 : |- ( ( ph -> ps ) -> ( ( ch -> th ) -> ( ( ps -> ch ) -> ( ph -> th ) ) ) )

theorem

set

[]

set.mm

imim12

Closed form of ~ imim12i and of ~ 3syl . (Contributed by BJ, 16-Jul-2019.)

jarr : |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) )

theorem

set

[]

set.mm

jarr

Elimination of a nested antecedent. Sometimes called "Syll-Simp" since it is a syllogism applied to ~ ax-1 ("Simplification"). (Contributed by Wolf Lammen, 9-May-2013.)