Theorem spcegf 2813

Description: Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)

Hypotheses

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

Assertion

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

Proof of Theorem spcegf

Step	Hyp	Ref	Expression
1		spcgf.2	. . 3 |- F/ x ps
2		spcgf.1	. . 3 |- F/_ x A
3	1, 2	spcegft 2809	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. V -> ( ps -> E. x ph ) ) )
4		spcgf.3	. 2 |- ( x = A -> ( ph <-> ps ) )
5	3, 4	mpg 1444	1 |- ( A e. V -> ( ps -> E. x ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   F/wnf 1453   E.wex 1485   e. wcel 2141   F/_wnfc 2299

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:  spcegv  2818  rspce  2829  euotd  4239  seq3f1olemstep  10457