Theorem frege72 38229

Description: If property A is hereditary in the R-sequence, if x has property A, and if y is a result of an application of the procedure R to x, then y has property A. Proposition 72 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege72.x	|- X e. U
frege72.y	|- Y e. V

Assertion

Ref	Expression
frege72	|- ( R hereditary A -> ( X e. A -> ( X R Y -> Y e. A ) ) )

Proof of Theorem frege72

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege72.y	. . . 4 |- Y e. V
2	1	frege58c 38215	. . 3 |- ( A. z ( X R z -> z e. A ) -> [. Y / z ]. ( X R z -> z e. A ) )
3		sbcim1 3482	. . . 4 |- ( [. Y / z ]. ( X R z -> z e. A ) -> ( [. Y / z ]. X R z -> [. Y / z ]. z e. A ) )
4		sbcbr2g 4710	. . . . . 6 |- ( Y e. V -> ( [. Y / z ]. X R z <-> X R [_ Y / z ]_ z ) )
5		csbvarg 4003	. . . . . . 7 |- ( Y e. V -> [_ Y / z ]_ z = Y )
6	5	breq2d 4665	. . . . . 6 |- ( Y e. V -> ( X R [_ Y / z ]_ z <-> X R Y ) )
7	4, 6	bitrd 268	. . . . 5 |- ( Y e. V -> ( [. Y / z ]. X R z <-> X R Y ) )
8	1, 7	ax-mp 5	. . . 4 |- ( [. Y / z ]. X R z <-> X R Y )
9		sbcel1v 3495	. . . 4 |- ( [. Y / z ]. z e. A <-> Y e. A )
10	3, 8, 9	3imtr3g 284	. . 3 |- ( [. Y / z ]. ( X R z -> z e. A ) -> ( X R Y -> Y e. A ) )
11	2, 10	syl 17	. 2 |- ( A. z ( X R z -> z e. A ) -> ( X R Y -> Y e. A ) )
12		frege72.x	. . 3 |- X e. U
13	12	frege71 38228	. 2 |- ( ( A. z ( X R z -> z e. A ) -> ( X R Y -> Y e. A ) ) -> ( R hereditary A -> ( X e. A -> ( X R Y -> Y e. A ) ) ) )
14	11, 13	ax-mp 5	1 |- ( R hereditary A -> ( X e. A -> ( X R Y -> Y e. A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   [.wsbc 3435   [_csb 3533   class class class wbr 4653   hereditary whe 38066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege52a 38151  ax-frege58b 38195

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-he 38067

This theorem is referenced by:  frege73  38230  frege74  38231