Theorem vtoclgaf 3172

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)

Hypotheses

Ref	Expression
vtoclgaf.1	|- F/_ x A
vtoclgaf.2	|- F/ x ps
vtoclgaf.3	|- ( x = A -> ( ph <-> ps ) )
vtoclgaf.4	|- ( x e. B -> ph )

Assertion

Ref	Expression
vtoclgaf	|- ( A e. B -> ps )

Distinct variable group:   x, B

Allowed substitution hints:   ph( x)   ps( x)   A( x)

Proof of Theorem vtoclgaf

Step	Hyp	Ref	Expression
1		vtoclgaf.1	. . 3 |- F/_ x A
2	1	nfel1 2635	. . . 4 |- F/ x A e. B
3		vtoclgaf.2	. . . 4 |- F/ x ps
4	2, 3	nfim 1921	. . 3 |- F/ x ( A e. B -> ps )
5		eleq1 2529	. . . 4 |- ( x = A -> ( x e. B <-> A e. B ) )
6		vtoclgaf.3	. . . 4 |- ( x = A -> ( ph <-> ps ) )
7	5, 6	imbi12d 320	. . 3 |- ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) )
8		vtoclgaf.4	. . 3 |- ( x e. B -> ph )
9	1, 4, 7, 8	vtoclgf 3165	. 2 |- ( A e. B -> ( A e. B -> ps ) )
10	9	pm2.43i 47	1 |- ( A e. B -> ps )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1395   F/wnf 1617   e. wcel 1819   F/_wnfc 2605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111

This theorem is referenced by:  vtoclga  3173  ssiun2s  4376  fvmptss  5965  fvmptf  5973  fmptco  6065  tfis  6688  inar1  9170  sumss  13558  fprodn0  13795  prmind2  14240  lss1d  17736  itg2splitlem  22281  dgrle  22766  cnlnadjlem5  27117  stoweidlem26  32011