Theorem spcimegft 2903

Description: A closed version of spcimegf 2906. (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 2833	. 2 |- ( A e. B -> A e. _V )
2		spcimgft.2	. . . . 5 |- F/_ x A
3	2	issetf 2829	. . . 4 |- ( A e. _V <-> E. x x = A )
4		exim 1652	. . . 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 1726	. . 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 1400   = wceq 1402   F/wnf 1513   E.wex 1545   e. wcel 2209   F/_wnfc 2379   _Vcvv 2821

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  spcegft  2904  spcimegf  2906  spcimedv  2911