Theorem spcegf 2820

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 2816	. 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 1451	1 |- ( A e. V -> ( ps -> E. x ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   F/wnf 1460   E.wex 1492   e. wcel 2148   F/_wnfc 2306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739

This theorem is referenced by:  spcegv  2825  rspce  2836  euotd  4253  seq3f1olemstep  10495