Theorem spcegft 2882

Description: A closed version of spcegf 2886. (Contributed by Jim Kingdon, 22-Jun-2018.)

Hypotheses

Ref	Expression
spcimgft.1	|- F/ x ps
spcimgft.2	|- F/_ x A

Assertion

Ref	Expression
spcegft	|- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( ps -> E. x ph ) ) )

Proof of Theorem spcegft

Step	Hyp	Ref	Expression
1		biimpr 130	. . . 4 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
2	1	imim2i 12	. . 3 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ps -> ph ) ) )
3	2	alimi 1501	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( ps -> ph ) ) )
4		spcimgft.1	. . 3 |- F/ x ps
5		spcimgft.2	. . 3 |- F/_ x A
6	4, 5	spcimegft 2881	. 2 |- ( A. x ( x = A -> ( ps -> ph ) ) -> ( A e. B -> ( ps -> E. x ph ) ) )
7	3, 6	syl 14	1 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( ps -> E. x ph ) ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   F/wnf 1506   E.wex 1538   e. wcel 2200   F/_wnfc 2359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  spcegf  2886