Theorem vtoclgaf 3141

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 2632	. . . 4 |- F/ x A e. B
3		vtoclgaf.2	. . . 4 |- F/ x ps
4	2, 3	nfim 1858	. . 3 |- F/ x ( A e. B -> ps )
5		eleq1 2526	. . . 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 3134	. 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 1370   F/wnf 1590   e. wcel 1758   F/_wnfc 2602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080

This theorem is referenced by:  vtoclga  3142  ssiun2s  4323  fvmptss  5892  fvmptf  5900  fmptco  5986  tfis  6576  inar1  9054  sumss  13320  prmind2  13893  lss1d  17168  itg2splitlem  21360  dgrle  21845  cnlnadjlem5  25628  fprodn0  27635  stoweidlem26  29970