Theorem spcimgft 2828

Description: A closed version of spcimgf 2832. (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 2763	. 2 |- ( A e. B -> A e. _V )
2		spcimgft.2	. . . . 5 |- F/_ x A
3	2	issetf 2759	. . . 4 |- ( A e. _V <-> E. x x = A )
4		exim 1610	. . . 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 1684	. . 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 1362   = wceq 1364   F/wnf 1471   E.wex 1503   e. wcel 2160   F/_wnfc 2319   _Vcvv 2752

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754

This theorem is referenced by:  spcgft  2829  spcimgf  2832  spcimdv  2836