Theorem spcimgft 2806

Description: A closed version of spcimgf 2810. (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 2741	. 2 |- ( A e. B -> A e. _V )
2		spcimgft.2	. . . . 5 |- F/_ x A
3	2	issetf 2737	. . . 4 |- ( A e. _V <-> E. x x = A )
4		exim 1592	. . . 4 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( E. x x = A -> E. x ( ph -> ps ) ) )
5	3, 4	syl5bi 151	. . 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 1666	. . 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 1346   = wceq 1348   F/wnf 1453   E.wex 1485   e. wcel 2141   F/_wnfc 2299   _Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  spcgft  2807  spcimgf  2810  spcimdv  2814