Theorem frege72 36602

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 36588	. . 3 |- ( A. z ( X R z -> z e. A ) -> [. Y / z ]. ( X R z -> z e. A ) )
3		sbcim1 3302	. . . 4 |- ( [. Y / z ]. ( X R z -> z e. A ) -> ( [. Y / z ]. X R z -> [. Y / z ]. z e. A ) )
4		sbcbr2g 4451	. . . . . 6 |- ( Y e. V -> ( [. Y / z ]. X R z <-> X R [_ Y / z ]_ z ) )
5		csbvarg 3796	. . . . . . 7 |- ( Y e. V -> [_ Y / z ]_ z = Y )
6	5	breq2d 4407	. . . . . 6 |- ( Y e. V -> ( X R [_ Y / z ]_ z <-> X R Y ) )
7	4, 6	bitrd 261	. . . . 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 3314	. . . 4 |- ( [. Y / z ]. z e. A <-> Y e. A )
10	3, 8, 9	3imtr3g 277	. . 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 36601	. 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 189   A.wal 1450   e. wcel 1904   [.wsbc 3255   [_csb 3349   class class class wbr 4395   hereditary whe 36438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-frege1 36457  ax-frege2 36458  ax-frege8 36476  ax-frege52a 36524  ax-frege58b 36568

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ifp 984  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-he 36439

This theorem is referenced by:  frege73  36603  frege74  36604