Theorem spcegf 2702

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 2698	. 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 1385	1 |- ( A e. V -> ( ps -> E. x ph ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   F/wnf 1394   E.wex 1426   e. wcel 1438   F/_wnfc 2215

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621

This theorem is referenced by:  spcegv  2707  rspce  2717  euotd  4081  seq3f1olemstep  9930