Theorem vtoclgaf 3030

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 2584	. . . 4 |- F/ x A e. B
3		vtoclgaf.2	. . . 4 |- F/ x ps
4	2, 3	nfim 1852	. . 3 |- F/ x ( A e. B -> ps )
5		eleq1 2498	. . . 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 3023	. 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 1369   F/wnf 1589   e. wcel 1756   F/_wnfc 2561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969

This theorem is referenced by:  vtoclga  3031  ssiun2s  4209  fvmptss  5777  fvmptf  5785  fmptco  5871  tfis  6460  inar1  8934  sumss  13193  prmind2  13766  lss1d  17021  itg2splitlem  21201  dgrle  21686  cnlnadjlem5  25426  fprodn0  27441  stoweidlem26  29774