Theorem spcimgft 3284

Description: A closed version of spcimgf 3286. (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 3212	. 2 |- ( A e. B -> A e. _V )
2		spcimgft.2	. . . . 5 |- F/_ x A
3	2	issetf 3208	. . . 4 |- ( A e. _V <-> E. x x = A )
4		exim 1761	. . . 4 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( E. x x = A -> E. x ( ph -> ps ) ) )
5	3, 4	syl5bi 232	. . 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 2098	. . 3 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )
8	5, 7	syl6ib 241	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. _V -> ( A. x ph -> ps ) ) )
9	1, 8	syl5 34	1 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B -> ( A. x ph -> ps ) ) )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  spcgft  3285  spcimgf  3286  spcimdv  3290  ss2iundf  37951  spcdvw  42426