Theorem spcimgft 2811

Description: A closed version of spcimgf 2815. (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 2746	. 2 |- ( A e. B -> A e. _V )
2		spcimgft.2	. . . . 5 |- F/_ x A
3	2	issetf 2742	. . . 4 |- ( A e. _V <-> E. x x = A )
4		exim 1597	. . . 4 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( E. x x = A -> E. x ( ph -> ps ) ) )
5	3, 4	biimtrid 152	. . 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 1671	. . 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 1351   = wceq 1353   F/wnf 1458   E.wex 1490   e. wcel 2146   F/_wnfc 2304   _Vcvv 2735

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737

This theorem is referenced by:  spcgft  2812  spcimgf  2815  spcimdv  2819