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Theorem spcegf 3187
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 1844 . . . 4  |-  F/ x  -.  ps
4 spcgf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54notbid 294 . . . 4  |-  ( x  =  A  ->  ( -.  ph  <->  -.  ps )
)
61, 3, 5spcgf 3186 . . 3  |-  ( A  e.  V  ->  ( A. x  -.  ph  ->  -. 
ps ) )
76con2d 115 . 2  |-  ( A  e.  V  ->  ( ps  ->  -.  A. x  -.  ph ) )
8 df-ex 1592 . 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 1372    = wceq 1374   E.wex 1591   F/wnf 1594    e. wcel 1762   F/_wnfc 2608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108
This theorem is referenced by:  spcegv  3192  rspce  3202  euotd  4741  rspcegf  30931  stoweidlem36  31291  stoweidlem46  31301  bnj607  32928  bnj1491  33067
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