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Theorem spcegf 3176
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 1887 . . . 4  |-  F/ x  -.  ps
4 spcgf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54notbid 294 . . . 4  |-  ( x  =  A  ->  ( -.  ph  <->  -.  ps )
)
61, 3, 5spcgf 3175 . . 3  |-  ( A  e.  V  ->  ( A. x  -.  ph  ->  -. 
ps ) )
76con2d 115 . 2  |-  ( A  e.  V  ->  ( ps  ->  -.  A. x  -.  ph ) )
8 df-ex 1600 . 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 1381    = wceq 1383   E.wex 1599   F/wnf 1603    e. wcel 1804   F/_wnfc 2591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097
This theorem is referenced by:  spcegv  3181  rspce  3191  euotd  4738  rspcegf  31352  stoweidlem36  31707  stoweidlem46  31717  bnj607  33707  bnj1491  33846
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