Theorem vtoclgaf 3169

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 2638	. . . 4 |- F/ x A e. B
3		vtoclgaf.2	. . . 4 |- F/ x ps
4	2, 3	nfim 1862	. . 3 |- F/ x ( A e. B -> ps )
5		eleq1 2532	. . . 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 3162	. 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 1374   F/wnf 1594   e. wcel 1762   F/_wnfc 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108

This theorem is referenced by:  vtoclga  3170  ssiun2s  4362  fvmptss  5949  fvmptf  5957  fmptco  6045  tfis  6660  inar1  9142  sumss  13495  prmind2  14076  lss1d  17385  itg2splitlem  21883  dgrle  22368  cnlnadjlem5  26652  fprodn0  28672  cncfiooicclem1  31187  stoweidlem26  31281