Theorem vtoclgaf 2706

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 2251	. . . 4 |- F/ x A e. B
3		vtoclgaf.2	. . . 4 |- F/ x ps
4	2, 3	nfim 1519	. . 3 |- F/ x ( A e. B -> ps )
5		eleq1 2162	. . . 4 |- ( x = A -> ( x e. B <-> A e. B ) )
6		vtoclgaf.3	. . . 4 |- ( x = A -> ( ph <-> ps ) )
7	5, 6	imbi12d 233	. . 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 2699	. 2 |- ( A e. B -> ( A e. B -> ps ) )
10	9	pm2.43i 49	1 |- ( A e. B -> ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1299   F/wnf 1404   e. wcel 1448   F/_wnfc 2227

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643

This theorem is referenced by:  vtoclga  2707  ssiun2s  3804  tfis  4435  fvmptf  5445  fmptco  5518  prmind2  11594