Theorem vtoclgaf 2920

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/_xA
vtoclgaf.2	|- F/xps
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/_xA
2	1	nfel1 2500	. . . 4 |- F/x A e. B
3		vtoclgaf.2	. . . 4 |- F/xps
4	2, 3	nfim 1813	. . 3 |- F/x(A e. B -> ps)
5		eleq1 2413	. . . 4 |- (x = A -> (x e. B <-> A e. B))
6		vtoclgaf.3	. . . 4 |- (x = A -> (ph <-> ps))
7	5, 6	imbi12d 311	. . 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 2914	. 2 |- (A e. B -> (A e. B -> ps))
10	9	pm2.43i 43	1 |- (A e. B -> ps)

Syntax hints:   -> wi 4   <-> wb 176   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  vtoclga  2921  ssiun2s  4011  fvmptss  5706  fvmptf  5723