Theorem spcimegft 2858

Description: A closed version of spcimegf 2861. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

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

Assertion

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

Proof of Theorem spcimegft

Step	Hyp	Ref	Expression
1		elex 2788	. 2 |- ( A e. B -> A e. _V )
2		spcimgft.2	. . . . 5 |- F/_ x A
3	2	issetf 2784	. . . 4 |- ( A e. _V <-> E. x x = A )
4		exim 1623	. . . 4 |- ( A. x ( x = A -> ( ps -> ph ) ) -> ( E. x x = A -> E. x ( ps -> ph ) ) )
5	3, 4	biimtrid 152	. . 3 |- ( A. x ( x = A -> ( ps -> ph ) ) -> ( A e. _V -> E. x ( ps -> ph ) ) )
6		spcimgft.1	. . . 4 |- F/ x ps
7	6	19.37-1 1698	. . 3 |- ( E. x ( ps -> ph ) -> ( ps -> E. x ph ) )
8	5, 7	syl6 33	. 2 |- ( A. x ( x = A -> ( ps -> ph ) ) -> ( A e. _V -> ( ps -> E. x ph ) ) )
9	1, 8	syl5 32	1 |- ( A. x ( x = A -> ( ps -> ph ) ) -> ( A e. B -> ( ps -> E. x ph ) ) )

Syntax hints:   -> wi 4   A.wal 1371   = wceq 1373   F/wnf 1484   E.wex 1516   e. wcel 2178   F/_wnfc 2337   _Vcvv 2776

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  spcegft  2859  spcimegf  2861  spcimedv  2866