Theorem spcegft 2809

Description: A closed version of spcegf 2813. (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 129	. . . 4 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
2	1	imim2i 12	. . 3 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ps -> ph ) ) )
3	2	alimi 1448	. 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 2808	. 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 104   A.wal 1346   = wceq 1348   F/wnf 1453   E.wex 1485   e. wcel 2141   F/_wnfc 2299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  spcegf  2813