Theorem spcimgft 2695

Description: A closed version of spcimgf 2699. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

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

Assertion

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

Proof of Theorem spcimgft

Step	Hyp	Ref	Expression
1		elex 2630	. 2 |- ( A e. B -> A e. _V )
2		spcimgft.2	. . . . 5 |- F/_ x A
3	2	issetf 2626	. . . 4 |- ( A e. _V <-> E. x x = A )
4		exim 1535	. . . 4 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( E. x x = A -> E. x ( ph -> ps ) ) )
5	3, 4	syl5bi 150	. . 3 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. _V -> E. x ( ph -> ps ) ) )
6		spcimgft.1	. . . 4 |- F/ x ps
7	6	19.36-1 1608	. . 3 |- ( E. x ( ph -> ps ) -> ( A. x ph -> ps ) )
8	5, 7	syl6 33	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. _V -> ( A. x ph -> ps ) ) )
9	1, 8	syl5 32	1 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B -> ( A. x ph -> ps ) ) )

Syntax hints:   -> wi 4   A.wal 1287   = wceq 1289   F/wnf 1394   E.wex 1426   e. wcel 1438   F/_wnfc 2215   _Vcvv 2619

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621

This theorem is referenced by:  spcgft  2696  spcimgf  2699  spcimdv  2703