Theorem spcegf 3289

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.1	. . . 4 |- F/_ x A
2		spcgf.2	. . . . 5 |- F/ x ps
3	2	nfn 1784	. . . 4 |- F/ x -. ps
4		spcgf.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
5	4	notbid 308	. . . 4 |- ( x = A -> ( -. ph <-> -. ps ) )
6	1, 3, 5	spcgf 3288	. . 3 |- ( A e. V -> ( A. x -. ph -> -. ps ) )
7	6	con2d 129	. 2 |- ( A e. V -> ( ps -> -. A. x -. ph ) )
8		df-ex 1705	. 2 |- ( E. x ph <-> -. A. x -. ph )
9	7, 8	syl6ibr 242	1 |- ( A e. V -> ( ps -> E. x ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  spcegv  3294  rspce  3304  euotd  4975  bnj607  30986  bnj1491  31125  rspcegf  39182  stoweidlem36  40253  stoweidlem46  40263