Theorem spcgft 2758

Description: A closed version of spcgf 2763. (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		bi1 117	. . . 4 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
2	1	imim2i 12	. . 3 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph -> ps ) ) )
3	2	alimi 1431	. 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 2757	. 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 1329   = wceq 1331   F/wnf 1436   e. wcel 1480   F/_wnfc 2266

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683

This theorem is referenced by:  spcgf  2763  rspct  2777