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 ⌀ |
|---|---|---|---|---|---|---|
impbii : |- ( ph <-> ps ) | theorem | set | [] | set.mm | impbii | Infer an equivalence from an implication and its converse. Inference associated with ~ impbi . (Contributed by NM, 29-Dec-1992.) |
impbidd : |- ( ph -> ( ps -> ( ch <-> th ) ) ) | theorem | set | [] | set.mm | impbidd | Deduce an equivalence from two implications. Double deduction associated with ~ impbi and ~ impbii . Deduction associated with ~ impbid . (Contributed by Rodolfo Medina, 12-Oct-2010.) |
impbid21d : |- ( ph -> ( ps -> ( ch <-> th ) ) ) | theorem | set | [] | set.mm | impbid21d | Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.) |
impbid : |- ( ph -> ( ps <-> ch ) ) | theorem | set | [] | set.mm | impbid | Deduce an equivalence from two implications. Deduction associated with ~ impbi and ~ impbii . (Contributed by NM, 24-Jan-1993.) Prove it from ~ impbid21d . (Revised by Wolf Lammen, 3-Nov-2012.) |
dfbi1 : |- ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) | theorem | set | [] | set.mm | dfbi1 | Relate the biconditional connective to primitive connectives. See ~ dfbi1ALT for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992.) |
dfbi1ALT : |- ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) | theorem | set | [] | set.mm | dfbi1ALT | Alternate proof of ~ dfbi1 . This proof, discovered by Gregory Bush on 8-Mar-2004, has several curious properties. First, it has only 17 steps directly from the axioms and ~ df-bi , compared to over 800 steps were the proof of ~ dfbi1 expanded into axioms. Second, step 2 demands only the property of "true"; any axiom (... |
biimp : |- ( ( ph <-> ps ) -> ( ph -> ps ) ) | theorem | set | [] | set.mm | biimp | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) |
biimpi : |- ( ph -> ps ) | theorem | set | [] | set.mm | biimpi | Infer an implication from a logical equivalence. Inference associated with ~ biimp . (Contributed by NM, 29-Dec-1992.) |
sylbi : |- ( ph -> ch ) | theorem | set | [] | set.mm | sylbi | A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 3-Jan-1993.) |
sylib : |- ( ph -> ch ) | theorem | set | [] | set.mm | sylib | A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993.) |
sylbb : |- ( ph -> ch ) | theorem | set | [] | set.mm | sylbb | A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019.) |
biimpr : |- ( ( ph <-> ps ) -> ( ps -> ph ) ) | theorem | set | [] | set.mm | biimpr | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) |
bicom1 : |- ( ( ph <-> ps ) -> ( ps <-> ph ) ) | theorem | set | [] | set.mm | bicom1 | Commutative law for the biconditional. (Contributed by Wolf Lammen, 10-Nov-2012.) |
bicom : |- ( ( ph <-> ps ) <-> ( ps <-> ph ) ) | theorem | set | [] | set.mm | bicom | Commutative law for the biconditional. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.) |
bicomd : |- ( ph -> ( ch <-> ps ) ) | theorem | set | [] | set.mm | bicomd | Commute two sides of a biconditional in a deduction. (Contributed by NM, 14-May-1993.) |
bicomi : |- ( ps <-> ph ) | theorem | set | [] | set.mm | bicomi | Inference from commutative law for logical equivalence. (Contributed by NM, 3-Jan-1993.) |
impbid1 : |- ( ph -> ( ps <-> ch ) ) | theorem | set | [] | set.mm | impbid1 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) |
impbid2 : |- ( ph -> ( ps <-> ch ) ) | theorem | set | [] | set.mm | impbid2 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) |
impcon4bid : |- ( ph -> ( ps <-> ch ) ) | theorem | set | [] | set.mm | impcon4bid | A variation on ~ impbid with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.) |
biimpri : |- ( ps -> ph ) | theorem | set | [] | set.mm | biimpri | Infer a converse implication from a logical equivalence. Inference associated with ~ biimpr . (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 16-Sep-2013.) |
biimpd : |- ( ph -> ( ps -> ch ) ) | theorem | set | [] | set.mm | biimpd | Deduce an implication from a logical equivalence. Deduction associated with ~ biimp and ~ biimpi . (Contributed by NM, 11-Jan-1993.) |
mpbi : |- ps | theorem | set | [] | set.mm | mpbi | An inference from a biconditional, related to modus ponens. (Contributed by NM, 11-May-1993.) |
mpbir : |- ph | theorem | set | [] | set.mm | mpbir | An inference from a biconditional, related to modus ponens. (Contributed by NM, 28-Dec-1992.) |
mpbid : |- ( ph -> ch ) | theorem | set | [] | set.mm | mpbid | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 21-Jun-1993.) |
mpbii : |- ( ph -> ch ) | theorem | set | [] | set.mm | mpbii | An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) |
sylibr : |- ( ph -> ch ) | theorem | set | [] | set.mm | sylibr | A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 3-Jan-1993.) |
sylbir : |- ( ph -> ch ) | theorem | set | [] | set.mm | sylbir | A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 3-Jan-1993.) |
sylbbr : |- ( ch -> ph ) | theorem | set | [] | set.mm | sylbbr | A mixed syllogism inference from two biconditionals. Note on the various syllogism-like statements in set.mm. The hypothetical syllogism ~ syl infers an implication from two implications (and there are ~ 3syl and ~ 4syl for chaining more inferences). There are four inferences inferring an implication from one implicati... |
sylbb1 : |- ( ps -> ch ) | theorem | set | [] | set.mm | sylbb1 | A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.) |
sylbb2 : |- ( ph -> ch ) | theorem | set | [] | set.mm | sylbb2 | A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.) |
sylibd : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | sylibd | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
sylbid : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | sylbid | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
mpbidi : |- ( th -> ( ph -> ch ) ) | theorem | set | [] | set.mm | mpbidi | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.) |
biimtrid : |- ( ch -> ( ph -> th ) ) | theorem | set | [] | set.mm | biimtrid | A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 12-Jan-1993.) |
biimtrrid : |- ( ch -> ( ph -> th ) ) | theorem | set | [] | set.mm | biimtrrid | A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.) |
imbitrid : |- ( ch -> ( ph -> th ) ) | theorem | set | [] | set.mm | imbitrid | A mixed syllogism inference. (Contributed by NM, 12-Jan-1993.) |
syl5ibcom : |- ( ph -> ( ch -> th ) ) | theorem | set | [] | set.mm | syl5ibcom | A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.) |
imbitrrid : |- ( ch -> ( ph -> ps ) ) | theorem | set | [] | set.mm | imbitrrid | A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) |
syl5ibrcom : |- ( ph -> ( ch -> ps ) ) | theorem | set | [] | set.mm | syl5ibrcom | A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.) |
biimprd : |- ( ph -> ( ch -> ps ) ) | theorem | set | [] | set.mm | biimprd | Deduce a converse implication from a logical equivalence. Deduction associated with ~ biimpr and ~ biimpri . (Contributed by NM, 11-Jan-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |
biimpcd : |- ( ps -> ( ph -> ch ) ) | theorem | set | [] | set.mm | biimpcd | Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |
biimprcd : |- ( ch -> ( ph -> ps ) ) | theorem | set | [] | set.mm | biimprcd | Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) |
imbitrdi : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | imbitrdi | A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.) |
imbitrrdi : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | imbitrrdi | A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. (Contributed by NM, 5-Aug-1993.) |
biimtrdi : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | biimtrdi | A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.) |
biimtrrdi : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | biimtrrdi | A mixed syllogism inference. (Contributed by NM, 18-May-1994.) |
syl7bi : |- ( ch -> ( th -> ( ph -> ta ) ) ) | theorem | set | [] | set.mm | syl7bi | A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-1993.) |
syl8ib : |- ( ph -> ( ps -> ( ch -> ta ) ) ) | theorem | set | [] | set.mm | syl8ib | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) |
mpbird : |- ( ph -> ps ) | theorem | set | [] | set.mm | mpbird | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
mpbiri : |- ( ph -> ps ) | theorem | set | [] | set.mm | mpbiri | An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) |
sylibrd : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | sylibrd | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
sylbird : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | sylbird | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
biid : |- ( ph <-> ph ) | theorem | set | [] | set.mm | biid | Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. This is part of Frege's eighth axiom per Proposition 54 of [Frege1879] p. 50; see also ~ eqid . (Contributed by NM, 2-Jun-1993.) |
biidd : |- ( ph -> ( ps <-> ps ) ) | theorem | set | [] | set.mm | biidd | Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.) |
pm5.1im : |- ( ph -> ( ps -> ( ph <-> ps ) ) ) | theorem | set | [] | set.mm | pm5.1im | Two propositions are equivalent if they are both true. Closed form of ~ 2th . Equivalent to a ~ biimp -like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version ` ( ph <-> ( ps <-> ( ph <-> ps ) ) ) ` . (Contributed by Wolf Lammen, 12-M... |
2th : |- ( ph <-> ps ) | theorem | set | [] | set.mm | 2th | Two truths are equivalent. (Contributed by NM, 18-Aug-1993.) |
2thd : |- ( ph -> ( ps <-> ch ) ) | theorem | set | [] | set.mm | 2thd | Two truths are equivalent. Deduction form. (Contributed by NM, 3-Jun-2012.) |
monothetic : |- ( ( ph -> ph ) <-> ( ps -> ps ) ) | theorem | set | [] | set.mm | monothetic | Two self-implications (see ~ id ) are equivalent. This theorem, rather trivial in our axiomatization, is (the biconditional form of) a standard axiom for monothetic BCI logic. This is the most general theorem of which ~ trujust is an instance. Relatedly, this would be the justification theorem if the definition of ` T.... |
ibi : |- ( ph -> ps ) | theorem | set | [] | set.mm | ibi | Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.) |
ibir : |- ( ph -> ps ) | theorem | set | [] | set.mm | ibir | Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.) |
ibd : |- ( ph -> ( ps -> ch ) ) | theorem | set | [] | set.mm | ibd | Deduction that converts a biconditional implied by one of its arguments, into an implication. Deduction associated with ~ ibi . (Contributed by NM, 26-Jun-2004.) |
pm5.74 : |- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph -> ps ) <-> ( ph -> ch ) ) ) | theorem | set | [] | set.mm | pm5.74 | Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.) |
pm5.74i : |- ( ( ph -> ps ) <-> ( ph -> ch ) ) | theorem | set | [] | set.mm | pm5.74i | Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.) |
pm5.74ri : |- ( ph -> ( ps <-> ch ) ) | theorem | set | [] | set.mm | pm5.74ri | Distribution of implication over biconditional (reverse inference form). (Contributed by NM, 1-Aug-1994.) |
pm5.74d : |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) ) | theorem | set | [] | set.mm | pm5.74d | Distribution of implication over biconditional (deduction form). (Contributed by NM, 21-Mar-1996.) |
pm5.74rd : |- ( ph -> ( ps -> ( ch <-> th ) ) ) | theorem | set | [] | set.mm | pm5.74rd | Distribution of implication over biconditional (deduction form). (Contributed by NM, 19-Mar-1997.) |
bitri : |- ( ph <-> ch ) | theorem | set | [] | set.mm | bitri | An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) |
bitr2i : |- ( ch <-> ph ) | theorem | set | [] | set.mm | bitr2i | An inference from transitive law for logical equivalence. (Contributed by NM, 12-Mar-1993.) |
bitr3i : |- ( ph <-> ch ) | theorem | set | [] | set.mm | bitr3i | An inference from transitive law for logical equivalence. (Contributed by NM, 2-Jun-1993.) |
bitr4i : |- ( ph <-> ch ) | theorem | set | [] | set.mm | bitr4i | An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.) |
bitrd : |- ( ph -> ( ps <-> th ) ) | theorem | set | [] | set.mm | bitrd | Register '<->' as an equality for its type (wff). $) $( $j equality 'wb' from 'biid' 'bicomi' 'bitri'; definition 'dfbi1' for 'wb'; $) ${ bitrd.1 $e |- ( ph -> ( ps <-> ch ) ) $. bitrd.2 $e |- ( ph -> ( ch <-> th ) ) $. $( Deduction form of ~ bitri . (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 14... |
bitr2d : |- ( ph -> ( th <-> ps ) ) | theorem | set | [] | set.mm | bitr2d | Deduction form of ~ bitr2i . (Contributed by NM, 9-Jun-2004.) |
bitr3d : |- ( ph -> ( ch <-> th ) ) | theorem | set | [] | set.mm | bitr3d | Deduction form of ~ bitr3i . (Contributed by NM, 14-May-1993.) |
bitr4d : |- ( ph -> ( ps <-> th ) ) | theorem | set | [] | set.mm | bitr4d | Deduction form of ~ bitr4i . (Contributed by NM, 30-Jun-1993.) |
bitrid : |- ( ch -> ( ph <-> th ) ) | theorem | set | [] | set.mm | bitrid | A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.) |
bitr2id : |- ( ch -> ( th <-> ph ) ) | theorem | set | [] | set.mm | bitr2id | A syllogism inference from two biconditionals. (Contributed by NM, 1-Aug-1993.) |
bitr3id : |- ( ch -> ( ph <-> th ) ) | theorem | set | [] | set.mm | bitr3id | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
bitr3di : |- ( ph -> ( ch <-> th ) ) | theorem | set | [] | set.mm | bitr3di | A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) |
bitrdi : |- ( ph -> ( ps <-> th ) ) | theorem | set | [] | set.mm | bitrdi | A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.) |
bitr2di : |- ( ph -> ( th <-> ps ) ) | theorem | set | [] | set.mm | bitr2di | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
bitr4di : |- ( ph -> ( ps <-> th ) ) | theorem | set | [] | set.mm | bitr4di | A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.) |
bitr4id : |- ( ph -> ( ps <-> th ) ) | theorem | set | [] | set.mm | bitr4id | A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) |
3imtr3i : |- ( ch -> th ) | theorem | set | [] | set.mm | 3imtr3i | A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.) |
3imtr4i : |- ( ch -> th ) | theorem | set | [] | set.mm | 3imtr4i | A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 3-Jan-1993.) |
3imtr3d : |- ( ph -> ( th -> ta ) ) | theorem | set | [] | set.mm | 3imtr3d | More general version of ~ 3imtr3i . Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.) |
3imtr4d : |- ( ph -> ( th -> ta ) ) | theorem | set | [] | set.mm | 3imtr4d | More general version of ~ 3imtr4i . Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.) |
3imtr3g : |- ( ph -> ( th -> ta ) ) | theorem | set | [] | set.mm | 3imtr3g | More general version of ~ 3imtr3i . Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) |
3imtr4g : |- ( ph -> ( th -> ta ) ) | theorem | set | [] | set.mm | 3imtr4g | More general version of ~ 3imtr4i . Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) |
3bitri : |- ( ph <-> th ) | theorem | set | [] | set.mm | 3bitri | A chained inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.) |
3bitrri : |- ( th <-> ph ) | theorem | set | [] | set.mm | 3bitrri | A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) |
3bitr2i : |- ( ph <-> th ) | theorem | set | [] | set.mm | 3bitr2i | A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) |
3bitr2ri : |- ( th <-> ph ) | theorem | set | [] | set.mm | 3bitr2ri | A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) |
3bitr3i : |- ( ch <-> th ) | theorem | set | [] | set.mm | 3bitr3i | A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.) |
3bitr3ri : |- ( th <-> ch ) | theorem | set | [] | set.mm | 3bitr3ri | A chained inference from transitive law for logical equivalence. (Contributed by NM, 21-Jun-1993.) |
3bitr4i : |- ( ch <-> th ) | theorem | set | [] | set.mm | 3bitr4i | A chained inference from transitive law for logical equivalence. This inference is frequently used to apply a definition to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) |
3bitr4ri : |- ( th <-> ch ) | theorem | set | [] | set.mm | 3bitr4ri | A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.) |
3bitrd : |- ( ph -> ( ps <-> ta ) ) | theorem | set | [] | set.mm | 3bitrd | Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.) |
3bitrrd : |- ( ph -> ( ta <-> ps ) ) | theorem | set | [] | set.mm | 3bitrrd | Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) |
3bitr2d : |- ( ph -> ( ps <-> ta ) ) | theorem | set | [] | set.mm | 3bitr2d | Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) |
3bitr2rd : |- ( ph -> ( ta <-> ps ) ) | theorem | set | [] | set.mm | 3bitr2rd | Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) |
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