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Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrege131d 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 )
 
Theoremfrege133d 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.

 
Theoremdfxor4 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 )
 ) )
 
Theoremdfxor5 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 )
 ) )
 
Theoremdf3or2 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 ) ) )
 
Theoremdf3an2 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 )
 ) )
 
Theoremnev 36407* Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
 |-  ( A  =/=  _V  <->  -.  A. x  x  e.  A )
 
Theoremdfss6 36408* Another definition of subclasshood. (Contributed by RP, 16-Apr-2020.)
 |-  ( A  C_  B  <->  -.  E. x ( x  e.  A  /\  -.  x  e.  B ) )
 
Theoremndisj 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 ) )
 
Theorem0pssin 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.

 
Theoremrp-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 ) )
 
Syntaxwhe 36412 The property of relation  R being hereditary in class  A.
 wff  R hereditary  A
 
Definitiondf-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 )
 
Theoremdfhe2 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 ) )
 
Theoremdfhe3 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 ) ) )
 
Theoremheeq12 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 ) )
 
Theoremheeq1 36417 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  ( R  =  S  ->  ( R hereditary  A  <->  S hereditary  A ) )
 
Theoremheeq2 36418 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
 |-  ( A  =  B  ->  ( R hereditary  A  <->  R hereditary  B ) )
 
Theoremsbcheg 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 ) )
 
Theoremhess 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 ) )
 
Theoremxphe 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
 
TheoremxpheOLD 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
 
Theorem0he 36423 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
 |-  (/) hereditary  A
 
Theorem0heALT 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
 
Theoremhe0 36425 Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)
 |-  A hereditary  (/)
 
Theoremunhe1 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 )
 
Theoremsnhesn 36427 Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
 |-  { <. A ,  A >. } hereditary  { B }
 
Theoremidhe 36428 The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
 |-  _I hereditary  A
 
Theorempsshepw 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
 
Theoremsshepw 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
 
Axiomax-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 ) )
 
Axiomax-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 ) ) )
 
Theoremrp-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 )
 ) )
 
Theoremrp-simp2 36434 Simplification of triple conjunction. Identical to simp2 1015. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ps )
 
Theoremrp-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 ) ) ) )
 
Theoremfrege3 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 ) ) ) )
 
Theoremrp-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 ) ) )
 
Theoremrp-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 ) ) )
 
Theoremrp-frege4g 36439 Deduction related to distribution. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ( ps 
 ->  ( ch  ->  th )
 ) )  ->  ( ph  ->  ( ( ps 
 ->  ch )  ->  ( ps  ->  th ) ) ) )
 
Theoremfrege4 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 ) ) ) )
 
Theoremfrege5 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 ) ) )
 
Theoremrp-7frege 36442 Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( th  ->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) ) )
 
Theoremrp-4frege 36443 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ( ( ps  ->  ph )  ->  ch ) )  ->  ( ph  ->  ch ) )
 
Theoremrp-6frege 36444 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
 |-  ( ph  ->  ( ( ps 
 ->  ( ( ch  ->  ps )  ->  th )
 )  ->  ( ps  ->  th ) ) )
 
Theoremrp-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 ) ) )
 
Theoremrp-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 )
 ) ) )
 
Theoremfrege6 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 ) ) ) )
 
Theoremaxfrege8 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 ) ) )
 
Theoremfrege7 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 ) ) ) )
 
Axiomax-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 ) ) )
 
Theoremfrege26 36451 Identical to idd 25. Proposition 26 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  ( ps  ->  ps ) )
 
Theoremfrege27 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 )
 
Theoremfrege9 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 ) ) )
 
Theoremfrege12 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 )
 ) ) )
 
Theoremfrege11 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 )
 )
 
Theoremfrege24 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 ) ) )
 
Theoremfrege16 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 ) ) ) ) )
 
Theoremfrege25 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 )
 ) ) )
 
Theoremfrege18 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 )
 ) ) )
 
Theoremfrege22 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 )
 ) ) ) ) )
 
Theoremfrege10 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 ) )
 
Theoremfrege17 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 ) ) ) )
 
Theoremfrege13 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 )
 ) ) )
 
Theoremfrege14 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 ) ) ) ) )
 
Theoremfrege19 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 )
 ) ) )
 
Theoremfrege23 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 ) ) ) ) )
 
Theoremfrege15 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 ) ) ) ) )
 
Theoremfrege21 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 )
 ) )
 
Theoremfrege20 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
 
Theoremaxfrege28 36470 Contraposition. Identical to con3 141. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  ps )  ->  ( -.  ps  ->  -.  ph ) )
 
Axiomax-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 ) )
 
Theoremfrege29 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 ) ) )
 
Theoremfrege30 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 ) ) )
 
Theoremaxfrege31 36474 Identical to notnot2 117. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.)
 |-  ( -.  -.  ph  ->  ph )
 
Axiomax-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 )
 
Theoremfrege32 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 ) ) )
 
Theoremfrege33 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 ) )
 
Theoremfrege34 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 ) ) )
 
Theoremfrege35 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 ) ) )
 
Theoremfrege36 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 ) )
 
Theoremfrege37 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 )
 )
 
Theoremfrege38 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 ) )
 
Theoremfrege39 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 ) )
 
Theoremfrege40 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 ) )
 
Theoremaxfrege41 36485 Identical to notnot1 127. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.)
 |-  ( ph  ->  -.  -.  ph )
 
Axiomax-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 )
 
Theoremfrege42 36487 Not not id 22. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  -.  -.  ( ph  ->  ph )
 
Theoremfrege43 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 )
 
Theoremfrege44 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 )
 )
 
Theoremfrege45 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 ) ) )
 
Theoremfrege46 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 )
 )
 
Theoremfrege47 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 ) ) )
 
Theoremfrege48 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 ) ) ) )
 
Theoremfrege49 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 ) ) )
 
Theoremfrege50 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 ) ) )
 
Theoremfrege51 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 ).

 
Theoremaxfrege52a 36497 Justification for ax-frege52a 36498. (Contributed by RP, 17-Apr-2020.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ph ,  th ,  ch )  -> if- ( ps ,  th ,  ch ) ) )
 
Axiomax-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 ) ) )
 
Theoremfrege52aid 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 ) )
 
Theoremfrege53aid 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 ) )
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