Theorem frege70 38227

Description: Lemma for frege72 38229. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege70.x	|- X e. V

Assertion

Ref	Expression
frege70	|- ( R hereditary A -> ( X e. A -> A. y ( X R y -> y e. A ) ) )

Distinct variable groups:   y, A   y, R   y, X

Allowed substitution hint:   V( y)

Proof of Theorem frege70

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffrege69 38226	. 2 |- ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) <-> R hereditary A )
2		frege70.x	. . . 4 |- X e. V
3	2	frege68c 38225	. . 3 |- ( ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) <-> R hereditary A ) -> ( R hereditary A -> [. X / x ]. ( x e. A -> A. y ( x R y -> y e. A ) ) ) )
4		sbcel1v 3495	. . . . 5 |- ( [. X / x ]. x e. A <-> X e. A )
5	4	biimpri 218	. . . 4 |- ( X e. A -> [. X / x ]. x e. A )
6		sbcim1 3482	. . . 4 |- ( [. X / x ]. ( x e. A -> A. y ( x R y -> y e. A ) ) -> ( [. X / x ]. x e. A -> [. X / x ]. A. y ( x R y -> y e. A ) ) )
7		sbcal 3485	. . . . 5 |- ( [. X / x ]. A. y ( x R y -> y e. A ) <-> A. y [. X / x ]. ( x R y -> y e. A ) )
8		sbcim1 3482	. . . . . . 7 |- ( [. X / x ]. ( x R y -> y e. A ) -> ( [. X / x ]. x R y -> [. X / x ]. y e. A ) )
9		sbcbr1g 4709	. . . . . . . . 9 |- ( X e. V -> ( [. X / x ]. x R y <-> [_ X / x ]_ x R y ) )
10	2, 9	ax-mp 5	. . . . . . . 8 |- ( [. X / x ]. x R y <-> [_ X / x ]_ x R y )
11		csbvarg 4003	. . . . . . . . . 10 |- ( X e. V -> [_ X / x ]_ x = X )
12	2, 11	ax-mp 5	. . . . . . . . 9 |- [_ X / x ]_ x = X
13	12	breq1i 4660	. . . . . . . 8 |- ( [_ X / x ]_ x R y <-> X R y )
14	10, 13	bitri 264	. . . . . . 7 |- ( [. X / x ]. x R y <-> X R y )
15		sbcg 3503	. . . . . . . 8 |- ( X e. V -> ( [. X / x ]. y e. A <-> y e. A ) )
16	2, 15	ax-mp 5	. . . . . . 7 |- ( [. X / x ]. y e. A <-> y e. A )
17	8, 14, 16	3imtr3g 284	. . . . . 6 |- ( [. X / x ]. ( x R y -> y e. A ) -> ( X R y -> y e. A ) )
18	17	alimi 1739	. . . . 5 |- ( A. y [. X / x ]. ( x R y -> y e. A ) -> A. y ( X R y -> y e. A ) )
19	7, 18	sylbi 207	. . . 4 |- ( [. X / x ]. A. y ( x R y -> y e. A ) -> A. y ( X R y -> y e. A ) )
20	5, 6, 19	syl56 36	. . 3 |- ( [. X / x ]. ( x e. A -> A. y ( x R y -> y e. A ) ) -> ( X e. A -> A. y ( X R y -> y e. A ) ) )
21	3, 20	syl6 35	. 2 |- ( ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) <-> R hereditary A ) -> ( R hereditary A -> ( X e. A -> A. y ( X R y -> y e. A ) ) ) )
22	1, 21	ax-mp 5	1 |- ( R hereditary A -> ( X e. A -> A. y ( X R y -> y e. A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   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:  frege71  38228