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Theorem spcegf 3048
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
StepHypRef Expression
1 spcgf.1 . . . 4  |-  F/_ x A
2 spcgf.2 . . . . 5  |-  F/ x ps
32nfn 1835 . . . 4  |-  F/ x  -.  ps
4 spcgf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54notbid 294 . . . 4  |-  ( x  =  A  ->  ( -.  ph  <->  -.  ps )
)
61, 3, 5spcgf 3047 . . 3  |-  ( A  e.  V  ->  ( A. x  -.  ph  ->  -. 
ps ) )
76con2d 115 . 2  |-  ( A  e.  V  ->  ( ps  ->  -.  A. x  -.  ph ) )
8 df-ex 1587 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
97, 8syl6ibr 227 1  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1367    = wceq 1369   E.wex 1586   F/wnf 1589    e. wcel 1756   F/_wnfc 2561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969
This theorem is referenced by:  spcegv  3053  rspce  3063  euotd  4587  rspcegf  29698  stoweidlem36  29784  stoweidlem46  29794  bnj607  31796  bnj1491  31935
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