Theorem spcgft 2803

Description: A closed version of spcgf 2808. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)

Hypotheses

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

Assertion

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

Proof of Theorem spcgft

Step	Hyp	Ref	Expression
1		biimp 117	. . . 4 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
2	1	imim2i 12	. . 3 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph -> ps ) ) )
3	2	alimi 1443	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( ph -> ps ) ) )
4		spcimgft.1	. . 3 |- F/ x ps
5		spcimgft.2	. . 3 |- F/_ x A
6	4, 5	spcimgft 2802	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B -> ( A. x ph -> ps ) ) )
7	3, 6	syl 14	1 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   F/wnf 1448   e. wcel 2136   F/_wnfc 2295

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  spcgf  2808  rspct  2823