Theorem spcegf 2935

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

Hypotheses

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

Assertion

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

Proof of Theorem spcegf

Step	Hyp	Ref	Expression
1		spcgf.1	. . . 4 |- F/_xA
2		spcgf.2	. . . . 5 |- F/xps
3	2	nfn 1793	. . . 4 |- F/x -. ps
4		spcgf.3	. . . . 5 |- (x = A -> (ph <-> ps))
5	4	notbid 285	. . . 4 |- (x = A -> (-. ph <-> -. ps))
6	1, 3, 5	spcgf 2934	. . 3 |- (A e. V -> (A.x -. ph -> -. ps))
7	6	con2d 107	. 2 |- (A e. V -> (ps -> -. A.x -. ph))
8		df-ex 1542	. 2 |- (E.xph <-> -. A.x -. ph)
9	7, 8	syl6ibr 218	1 |- (A e. V -> (ps -> E.xph))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861

This theorem is referenced by:  spcegv  2940  rspce  2950