P. 365 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	frege131d 36401	If F is a function and A contains all elements of U and all elements before or after those elements of U in the transitive closure of F, then the image under F of A is a subclass of A. Similar to Proposition 131 of [Frege1879] p. 85. Compare with frege131 36635. (Contributed by RP, 17-Jul-2020.)
|- ( ph -> F e. _V )   &    |- ( ph -> A = ( U u. ( ( `' ( t+ ` F ) " U ) u. ( ( t+ ` F ) " U ) ) ) )   &    |- ( ph -> Fun F )   =>    |- ( ph -> ( F " A ) C_ A )
Theorem	frege133d 36402	If F is a function and A and B both follow X in the transitive closure of F, then (for distinct A and B) either A follows B or B follows A in the transitive closure of F (or both if it loops). Similar to Proposition 133 of [Frege1879] p. 86. Compare with frege133 36637. (Contributed by RP, 18-Jul-2020.)
|- ( ph -> F e. _V )   &    |- ( ph -> X ( t+ ` F ) A )   &    |- ( ph -> X ( t+ ` F ) B )   &    |- ( ph -> Fun F )   =>    |- ( ph -> ( A ( t+ ` F ) B \/ A = B \/ B ( t+ ` F ) A ) )
21.25.3  Propositions from _Begriffsschrift_ In 1879, Frege introduced notation for documenting formal reasoning about propositions (and classes) which covered elements of propositional logic, predicate calculus and reasoning about relations. However, due to the pitfalls of naive set theory, adapting this work for inclusion in set.mm required dividing statements about propositions from those about classes and identifying when a restriction to sets is required. For an overview comparing the details of Frege's two-dimensional notation and that used in set.mm, see mmfrege.html. See ru 3278 for discussion of an example of a class that is not a set. Numbered propositions from [Frege1879]. ax-frege1 36431, ax-frege2 36432, ax-frege8 36450, ax-frege28 36471, ax-frege31 36475, ax-frege41 36486, frege52 (see ax-frege52a 36498, frege52b 36530, and ax-frege52c 36529 for translations), frege54 (see ax-frege54a 36503, frege54b 36534 and ax-frege54c 36533 for translations) and frege58 (see ax-frege58a 36516, ax-frege58b 36542 and frege58c 36562 for translations) are considered "core" or axioms. However, at least ax-frege8 36450 can be derived from ax-frege1 36431 and ax-frege2 36432, see axfrege8 36448. Frege introduced implication, negation and the universal qualifier as primitives and did not in the numbered propositions use other logical connectives other than equivalence introduced in ax-frege52a 36498, frege52b 36530, and ax-frege52c 36529. In dffrege69 36573, Frege introduced R hereditary A to say that relation R, when restricted to operate on elements of class A, will only have elements of class A in its domain; see df-he 36413 for a definition in terms of image and subset. In dffrege76 36580, Frege introduced notation for the concept of two sets related by the transitive closure of a relation, for which we write X ( t+ ` R ) Y, which requires R to also be a set. In dffrege99 36603, Frege introduced notation for the concept of two sets either identical or related by the transitive closure of a relation, for which we write X ( ( t+ ` R ) u. _I ) Y, which is a superclass of sets related by the reflexive-transitive relation X ( t* ` R ) Y. Finally, in dffrege115 36619, Frege introduced notation for the concept of a relation having the property elements in its domain pair up with only one element each in its range, for which we write Fun `' `' R (to ignore any non-relational content of the class R). Frege did this without the expressing concept of a relation (or its transitive closure) as a class, and needed to invent conventions for discussing indeterminate propositions with two slots free and how to recognize which of the slots was domain and which was range. See mmfrege.html for details. English translations for specific propositions lifted in part from a translation by Stefan Bauer-Mengelberg as reprinted in From Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931. An attempt to align these propositions in the larger set.mm database has also been made. See frege77d 36383 for an example.
21.25.3.1  _Begriffsschrift_ Chapter I Section 2 introduces the turnstile |- which turns an idea which may be true ph into an assertion that it does hold true |- ph. Section 5 introduces implication, ( ph -> ps ). Section 6 introduces the single rule of interference relied upon, modus ponens ax-mp 5. Section 7 introduces negation and with in synonyms for or ( -. ph -> ps ), and -. ( ph -> -. ps ), and two for exclusive-or corresponding to df-or 376, df-an 377, dfxor4 36403, dfxor5 36404. Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biimplication ( ph <-> ps ) or class equality A = B in this adaptation. Section 10 introduces "truth functions" for one or two variables in equally troubling notation, as the arguments may be understood to be logical predicates or collections. Here f( ph) is interpreted to mean if- ( ph , ps , ch ) where the content of the "function" is specified by the latter two argments or logical equivalent, while g( A) is read as A e. G and h( A , B) as A H B. This necessarily introduces a need for set theory as both A e. G and A H B cannot hold unless A is a set. (Also B.) Section 11 introduces notation for generality, but there is no standard notation for generality when the variable is a proposition because it was realized after Frege that the universe of all possible propositions includes paradoxical constructions leading to the failure of naive set theory. So adopting f( ph) as if- ( ph , ps , ch ) would result in the translation of A. ph f ( ph) as ( ps /\ ch ). For collections, we must generalize over set variables or run into the same problems; this leads to A. A g( A) being translated as A. a a e. G and so forth. Under this interpreation the text of section 11 gives us sp 1948 (or simpl 463 and simpr 467 and anifp 1443 in the propositional case) and statments similar to cbvalivw 1861, ax-gen 1680, alrimiv 1784, and alrimdv 1786. These last four introduce a generality and have no useful definition in terms of propositional variables. Section 12 introduces some combinations of primitive symbols and their human language counterparts. Using class notation, these can also be expressed without dummy variables. All are A, A. x x e. A, -. E. x -. x e. A alex 1709, A = _V eqv 3760; Some are not B, -. A. x x e. B, E. x -. x e. B exnal 1710, B C. _V pssv 3816, B =/= _V nev 36407; There are no C, A. x -. x e. C, -. E. x x e. C alnex 1676, C = (/) eq0 3759; There exist D, -. A. x -. x e. D, E. x x e. D df-ex 1675, (/) C. D 0pss 3814, D =/= (/) n0 3753. Notation for relations between expressions also can be written in various ways. All E are P, A. x ( x e. E -> x e. P ), -. E. x ( x e. E /\ -. x e. P ) dfss6 36408, E = ( E i^i P ) df-ss 3430, E C_ P dfss2 3433; No F are P, A. x ( x e. F -> -. x e. P ), -. E. x ( x e. F /\ x e. P ) alinexa 1724, ( F i^i P ) = (/) disj1 3819; Some G are not P, -. A. x ( x e. G -> x e. P ), E. x ( x e. G /\ -. x e. P ) exanali 1732, ( G i^i P ) C. G nssinpss 3687, -. G C_ P nss 3502; Some H are P, -. A. x ( x e. H -> -. x e. P ), E. x ( x e. H /\ x e. P ) bj-exnalimn 31266, (/) C. ( H i^i P ) 0pssin 36410, ( H i^i P ) =/= (/) ndisj 36409.
Theorem	dfxor4 36403	Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
|- ( ( ph \/_ ps ) <-> -. ( ( -. ph -> ps ) -> -. ( ph -> -. ps ) ) )
Theorem	dfxor5 36404	Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
|- ( ( ph \/_ ps ) <-> -. ( ( ph -> -. ps ) -> -. ( -. ph -> ps ) ) )
Theorem	df3or2 36405	Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.)
|- ( ( ph \/ ps \/ ch ) <-> ( -. ph -> ( -. ps -> ch ) ) )
Theorem	df3an2 36406	Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.)
|- ( ( ph /\ ps /\ ch ) <-> -. ( ph -> ( ps -> -. ch ) ) )
Theorem	nev 36407*	Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
|- ( A =/= _V <-> -. A. x x e. A )
Theorem	dfss6 36408*	Another definition of subclasshood. (Contributed by RP, 16-Apr-2020.)
|- ( A C_ B <-> -. E. x ( x e. A /\ -. x e. B ) )
Theorem	ndisj 36409*	Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
|- ( ( A i^i B ) =/= (/) <-> E. x ( x e. A /\ x e. B ) )
Theorem	0pssin 36410*	Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
|- ( (/) C. ( A i^i B ) <-> E. x ( x e. A /\ x e. B ) )
21.25.3.2  _Begriffsschrift_ Notation hints The statement R hereditary A means relation R is hereditary (in the sense of Frege) in the class A or ( R " A ) C_ A. The former is only a slight reduction in the number of symbols, but this reduces the number of floating hypotheses needed to be checked. As Frege wasn't using the language of classes or sets, this naturally differs from the set-theoretic notion that a set is hereditary in a property: that all of its elements have a property and all of their elements have the property and so-on.
Theorem	rp-imass 36411	If the R-image of a class A is a subclass of B, then the restriction of R to A is a subset of the Cartesian product of A and B. (Contributed by Richard Penner, 24-Dec-2019.)
|- ( ( R " A ) C_ B <-> ( R |` A ) C_ ( A X. B ) )
Syntax	whe 36412	The property of relation R being hereditary in class A.
wff R hereditary A
Definition	df-he 36413	The property of relation R being hereditary in class A. (Contributed by RP, 27-Mar-2020.)
|- ( R hereditary A <-> ( R " A ) C_ A )
Theorem	dfhe2 36414	The property of relation R being hereditary in class A. (Contributed by RP, 27-Mar-2020.)
|- ( R hereditary A <-> ( R |` A ) C_ ( A X. A ) )
Theorem	dfhe3 36415*	The property of relation R being hereditary in class A. (Contributed by RP, 27-Mar-2020.)
|- ( R hereditary A <-> A. x ( x e. A -> A. y ( x R y -> y e. A ) ) )
Theorem	heeq12 36416	Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
|- ( ( R = S /\ A = B ) -> ( R hereditary A <-> S hereditary B ) )
Theorem	heeq1 36417	Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
|- ( R = S -> ( R hereditary A <-> S hereditary A ) )
Theorem	heeq2 36418	Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
|- ( A = B -> ( R hereditary A <-> R hereditary B ) )
Theorem	sbcheg 36419	Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.)
|- ( A e. V -> ( [. A / x ]. B hereditary C <-> [_ A / x ]_ B hereditary [_ A / x ]_ C ) )
Theorem	hess 36420	Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
|- ( S C_ R -> ( R hereditary A -> S hereditary A ) )
Theorem	xphe 36421	Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) (Proof shortened by OpenAI, 3-Jul-2020.)
|- ( A X. B ) hereditary B
Theorem	xpheOLD 36422	Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) Obsolete version of xphe 36421 as of 3-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.)
|- ( A X. B ) hereditary B
Theorem	0he 36423	The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
|- (/) hereditary A
Theorem	0heALT 36424	The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
|- (/) hereditary A
Theorem	he0 36425	Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)
|- A hereditary (/)
Theorem	unhe1 36426	The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.)
|- ( ( R hereditary A /\ S hereditary A ) -> ( R u. S ) hereditary A )
Theorem	snhesn 36427	Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
|- { <. A , A >. } hereditary { B }
Theorem	idhe 36428	The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
|- _I hereditary A
Theorem	psshepw 36429	The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
|- `' [ C.] hereditary ~P A
Theorem	sshepw 36430	The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
|- ( `' [ C.] u. _I ) hereditary ~P A
21.25.3.3  _Begriffsschrift_ Chapter II Implication
Axiom	ax-frege1 36431	The case in which ph is denied, ps is affirmed, and ph is affirmed is excluded. This is evident since ph cannot at the same time be denied and affirmed. Axiom 1 of [Frege1879] p. 26. Identical to ax-1 6. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
|- ( ph -> ( ps -> ph ) )
Axiom	ax-frege2 36432	If a proposition ch is a necessary consequence of two propositions ps and ph and one of those, ps, is in turn a necessary consequence of the other, ph, then the proposition ch is a necessary consequence of the latter one, ph, alone. Axiom 2 of [Frege1879] p. 26. Identical to ax-2 7. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
Theorem	rp-simp2-frege 36433	Simplification of triple conjunction. Compare with simp2 1015. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ph -> ( ps -> ( ch -> ps ) ) )
Theorem	rp-simp2 36434	Simplification of triple conjunction. Identical to simp2 1015. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph /\ ps /\ ch ) -> ps )
Theorem	rp-frege3g 36435	Add antecedent to ax-frege2 36432. More general statement than frege3 36436. Like ax-frege2 36432, it is essentially a closed form of mpd 15, however it has an extra antecedent. It would be more natural to prove from a1i 11 and ax-frege2 36432 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ph -> ( ( ps -> ( ch -> th ) ) -> ( ( ps -> ch ) -> ( ps -> th ) ) ) )
Theorem	frege3 36436	Add antecedent to ax-frege2 36432. Special case of rp-frege3g 36435. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ps ) -> ( ( ch -> ( ph -> ps ) ) -> ( ( ch -> ph ) -> ( ch -> ps ) ) ) )
Theorem	rp-misc1-frege 36437	Double-use of ax-frege2 36432. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ( ph -> ( ps -> ch ) ) -> ( ph -> ps ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) )
Theorem	rp-frege24 36438	Introducing an embedded antecedent. Alternate proof for frege24 36456. Closed form for a1d 26. (Contributed by RP, 24-Dec-2019.)
|- ( ( ph -> ps ) -> ( ph -> ( ch -> ps ) ) )
Theorem	rp-frege4g 36439	Deduction related to distribution. (Contributed by RP, 24-Dec-2019.)
|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) )
Theorem	frege4 36440	Special case of closed form of a2d 29. Special case of rp-frege4g 36439. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ( ph -> ps ) -> ( ch -> ( ph -> ps ) ) ) -> ( ( ph -> ps ) -> ( ( ch -> ph ) -> ( ch -> ps ) ) ) )
Theorem	frege5 36441	A closed form of syl 17. Identical to imim2 55. Theorem *2.05 of [WhiteheadRussell] p. 100. Proposition 5 of [Frege1879] p. 32. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ps ) -> ( ( ch -> ph ) -> ( ch -> ps ) ) )
Theorem	rp-7frege 36442	Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.)
|- ( ( ph -> ( ps -> ch ) ) -> ( th -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
Theorem	rp-4frege 36443	Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
|- ( ( ph -> ( ( ps -> ph ) -> ch ) ) -> ( ph -> ch ) )
Theorem	rp-6frege 36444	Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
|- ( ph -> ( ( ps -> ( ( ch -> ps ) -> th ) ) -> ( ps -> th ) ) )
Theorem	rp-8frege 36445	Eliminate antecedent when it is implied by previous antecedent. (Contributed by RP, 24-Dec-2019.)
|- ( ( ph -> ( ps -> ( ( ch -> ps ) -> th ) ) ) -> ( ph -> ( ps -> th ) ) )
Theorem	rp-frege25 36446	Closed form for a1dd 47. Alternate route to Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )
Theorem	frege6 36447	A closed form of imim2d 54 which is a deduction adding nested antecedents. Proposition 6 of [Frege1879] p. 33. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ph -> ( ( th -> ps ) -> ( th -> ch ) ) ) )
Theorem	axfrege8 36448	Swap antecedents. Identical to pm2.04 85. This demonstrates that Axiom 8 of [Frege1879] p. 35 is redundant. Proof follows closely proof of pm2.04 85 in http://us.metamath.org/mmsolitaire/pmproofs.txt, but in the style of Frege's 1879 work. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
Theorem	frege7 36449	A closed form of syl6 34. The first antecedent is used to replace the consequent of the second antecedent. Proposition 7 of [Frege1879] p. 34. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ps ) -> ( ( ch -> ( th -> ph ) ) -> ( ch -> ( th -> ps ) ) ) )
Axiom	ax-frege8 36450	Swap antecedents. If two conditions have a proposition as a consequence, their order is immaterial. Third axiom of Frege's 1879 work but identical to pm2.04 85 which can be proved from only ax-mp 5, ax-frege1 36431, and ax-frege2 36432. (Redundant) Axiom 8 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
Theorem	frege26 36451	Identical to idd 25. Proposition 26 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ph -> ( ps -> ps ) )
Theorem	frege27 36452	We cannot (at the same time) affirm ph and deny ph. Identical to id 22. Proposition 27 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ph -> ph )
Theorem	frege9 36453	Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 36441 only in an unessential way. Identical to imim1 79. Proposition 9 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )
Theorem	frege12 36454	A closed form of com23 81. Proposition 12 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ph -> ( ch -> ( ps -> th ) ) ) )
Theorem	frege11 36455	Elimination of a nested antecedent as a partial converse of ja 166. If the proposition that ps takes place or ph does not is a sufficient condition for ch, then ps by itself is a sufficient condition for ch. Identical to jarr 101. Proposition 11 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) )
Theorem	frege24 36456	Closed form for a1d 26. Deduction introducing an embedded antecedent. Identical to rp-frege24 36438 which was proved without relying on ax-frege8 36450. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ps ) -> ( ph -> ( ch -> ps ) ) )
Theorem	frege16 36457	A closed form of com34 86. Proposition 16 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) -> ( ph -> ( ps -> ( th -> ( ch -> ta ) ) ) ) )
Theorem	frege25 36458	Closed form for a1dd 47. Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )
Theorem	frege18 36459	Closed form of a syllogism followed by a swap of antecedents. Proposition 18 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ph ) -> ( ps -> ( th -> ch ) ) ) )
Theorem	frege22 36460	A closed form of com45 92. Proposition 22 of [Frege1879] p. 41. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) -> ( ph -> ( ps -> ( ch -> ( ta -> ( th -> et ) ) ) ) ) )
Theorem	frege10 36461	Result commuting antecedents within an antecedent. Proposition 10 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ( ph -> ( ps -> ch ) ) -> th ) -> ( ( ps -> ( ph -> ch ) ) -> th ) )
Theorem	frege17 36462	A closed form of com3l 84. Proposition 17 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ps -> ( ch -> ( ph -> th ) ) ) )
Theorem	frege13 36463	A closed form of com3r 82. Proposition 13 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ch -> ( ph -> ( ps -> th ) ) ) )
Theorem	frege14 36464	Closed form of a deduction based on com3r 82. Proposition 14 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) -> ( ph -> ( th -> ( ps -> ( ch -> ta ) ) ) ) )
Theorem	frege19 36465	A closed form of syl6 34. Proposition 19 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ( ch -> th ) -> ( ph -> ( ps -> th ) ) ) )
Theorem	frege23 36466	Syllogism followed by rotation of three antecedents. Proposition 23 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ta -> ph ) -> ( ps -> ( ch -> ( ta -> th ) ) ) ) )
Theorem	frege15 36467	A closed form of com4r 89. Proposition 15 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) -> ( th -> ( ph -> ( ps -> ( ch -> ta ) ) ) ) )
Theorem	frege21 36468	Replace antecedent in antecedent. Proposition 21 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ( ph -> ps ) -> ch ) -> ( ( ph -> th ) -> ( ( th -> ps ) -> ch ) ) )
Theorem	frege20 36469	A closed form of syl8 72. Proposition 20 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( th -> ta ) -> ( ph -> ( ps -> ( ch -> ta ) ) ) ) )
21.25.3.4  _Begriffsschrift_ Chapter II Implication and Negation
Theorem	axfrege28 36470	Contraposition. Identical to con3 141. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.)
|- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )
Axiom	ax-frege28 36471	Contraposition. Identical to con3 141. Theorem *2.16 of [WhiteheadRussell] p. 103. Axiom 28 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
|- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )
Theorem	frege29 36472	Closed form of con3d 140. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ph -> ( -. ch -> -. ps ) ) )
Theorem	frege30 36473	Commuted, closed form of con3d 140. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( -. ch -> -. ph ) ) )
Theorem	axfrege31 36474	Identical to notnot2 117. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.)
|- ( -. -. ph -> ph )
Axiom	ax-frege31 36475	ph cannot be denied and (at the same time ) -. -. ph affirmed. Duplex negatio affirmat. The denial of the denial is affirmation. Identical to notnot2 117. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
|- ( -. -. ph -> ph )
Theorem	frege32 36476	Deduce con1 133 from con3 141. Proposition 32 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ( -. ph -> ps ) -> ( -. ps -> -. -. ph ) ) -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
Theorem	frege33 36477	If ph or ps takes place, then ps or ph takes place. Identical to con1 133. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( -. ph -> ps ) -> ( -. ps -> ph ) )
Theorem	frege34 36478	If as a conseqence of the occurence of the circumstance ph, when the obstacle ps is removed, ch takes place, then from the circumstance that ch does not take place while ph occurs the occurence of the obstacle ps can be inferred. Closed form of con1d 129. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( -. ps -> ch ) ) -> ( ph -> ( -. ch -> ps ) ) )
Theorem	frege35 36479	Commuted, closed form of con1d 129. Proposition 35 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( -. ps -> ch ) ) -> ( -. ch -> ( ph -> ps ) ) )
Theorem	frege36 36480	The case in which ps is denied, -. ph is affirmed, and ph is affirmed does not occur. If ph occurs, then (at least) one of the two, ph or ps, takes place (no matter what ps might be). Identical to pm2.24 113. Proposition 36 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ph -> ( -. ph -> ps ) )
Theorem	frege37 36481	If ch is a necessary consequence of the occurrence of ps or ph, then ch is a necessary consequence of ph alone. Similar to a closed form of orcs 400. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ( -. ph -> ps ) -> ch ) -> ( ph -> ch ) )
Theorem	frege38 36482	Identical to pm2.21 112. Proposition 38 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( -. ph -> ( ph -> ps ) )
Theorem	frege39 36483	Syllogism between pm2.18 114 and pm2.24 113. Proposition 39 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( -. ph -> ph ) -> ( -. ph -> ps ) )
Theorem	frege40 36484	Anything implies pm2.18 114. Proposition 40 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( -. ph -> ( ( -. ps -> ps ) -> ps ) )
Theorem	axfrege41 36485	Identical to notnot1 127. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.)
|- ( ph -> -. -. ph )
Axiom	ax-frege41 36486	The affirmation of ph denies the denial of ph. Identical to notnot1 127. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
|- ( ph -> -. -. ph )
Theorem	frege42 36487	Not not id 22. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- -. -. ( ph -> ph )
Theorem	frege43 36488	If there is a choice only between ph and ph, then ph takes place. Identical to pm2.18 114. Proposition 43 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( -. ph -> ph ) -> ph )
Theorem	frege44 36489	Similar to a commuted pm2.62 415. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( -. ph -> ps ) -> ( ( ps -> ph ) -> ph ) )
Theorem	frege45 36490	Deduce pm2.6 175 from con1 133. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) -> ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) )
Theorem	frege46 36491	If ps holds when ph occurs as well as when ph does not occur, then ps holds. If ps or ph occurs and if the occurences of ph has ps as a necessary consequence, then ps takes place. Identical to pm2.6 175. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) )
Theorem	frege47 36492	Deduce consequence follows from either path implied by a disjunction. If ph, as well as ps is sufficient condition for ch and ps or ph takes place, then the proposition ch holds. Proposition 47 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( -. ph -> ps ) -> ( ( ps -> ch ) -> ( ( ph -> ch ) -> ch ) ) )
Theorem	frege48 36493	Closed form of syllogism with internal disjunction. If ph is a sufficient condition for the occurence of ch or ps and if ch, as well as ps, is a sufficient condition for th, then ph is a sufficient condition for th. See application in frege101 36605. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( -. ps -> ch ) ) -> ( ( ch -> th ) -> ( ( ps -> th ) -> ( ph -> th ) ) ) )
Theorem	frege49 36494	Closed form of deduction with disjunction. Proposition 49 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( -. ph -> ps ) -> ( ( ph -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
Theorem	frege50 36495	Closed form of jaoi 385. Proposition 50 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ps ) -> ( ( ch -> ps ) -> ( ( -. ph -> ch ) -> ps ) ) )
Theorem	frege51 36496	Compare with jaod 386. Proposition 51 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ch ) -> ( ph -> ( ( -. ps -> th ) -> ch ) ) ) )
21.25.3.5  _Begriffsschrift_ Chapter II with logical equivalence Here we leverage df-ifp 1436 to partition a wff into two that are disjoint with the selector wff. Thus if we are given |- ( ph <-> if- ( ps , ch , th ) ) then we replace the concept (illegal in our notation ) ( ph ` ps ) with if- ( ps , ch , th ) to reason about the values of the "function." Likewise, we replace the similarly illegal concept A. ps ph with ( ch /\ th ).
Theorem	axfrege52a 36497	Justification for ax-frege52a 36498. (Contributed by RP, 17-Apr-2020.)
|- ( ( ph <-> ps ) -> (if- ( ph , th , ch ) -> if- ( ps , th , ch ) ) )
Axiom	ax-frege52a 36498	The case when the content of ph is identical with the content of ps and in which a proposition controlled by an element for which we substitute the content of ph is affirmed ( in this specific case the identity logical funtion ) and the same proposition, this time where we subsituted the content of ps, is denied does not take place. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
|- ( ( ph <-> ps ) -> (if- ( ph , th , ch ) -> if- ( ps , th , ch ) ) )
Theorem	frege52aid 36499	The case when the content of ph is identical with the content of ps and in which ph is affirmed and ps is denied does not take place. Identical to biimp 198. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ( ph <-> ps ) -> ( ph -> ps ) )
Theorem	frege53aid 36500	Specialization of frege53a 36501. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
|- ( ph -> ( ( ph <-> ps ) -> ps ) )