Theorem spcimegf 2679

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 2676	. 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 1380	1 |- ( A e. V -> ( ps -> E. x ph ) )

Syntax hints:   -> wi 4   = wceq 1284   F/wnf 1389   E.wex 1421   e. wcel 1433   F/_wnfc 2206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by: (None)