Theorem spcimegf 2634

Description: Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimgf.1	|- F/_ x A
spcimgf.2	|- F/ x ps
spcimegf.3	|- ( x = A -> ( ps -> ph ) )

Assertion

Ref	Expression
spcimegf	|- ( A e. V -> ( ps -> E. x ph ) )

Proof of Theorem spcimegf

Step	Hyp	Ref	Expression
1		spcimgf.2	. . 3 |- F/ x ps
2		spcimgf.1	. . 3 |- F/_ x A
3	1, 2	spcimegft 2631	. 2 |- ( A. x ( x = A -> ( ps -> ph ) ) -> ( A e. V -> ( ps -> E. x ph ) ) )
4		spcimegf.3	. 2 |- ( x = A -> ( ps -> ph ) )
5	3, 4	mpg 1340	1 |- ( A e. V -> ( ps -> E. x ph ) )

Syntax hints:   -> wi 4   = wceq 1243   F/wnf 1349   E.wex 1381   e. wcel 1393   F/_wnfc 2165

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by: (None)