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Theorem spcegf 2976
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 1801 . . . 4  |-  F/ x  -.  ps
4 spcgf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54notbid 286 . . . 4  |-  ( x  =  A  ->  ( -.  ph  <->  -.  ps )
)
61, 3, 5spcgf 2975 . . 3  |-  ( A  e.  V  ->  ( A. x  -.  ph  ->  -. 
ps ) )
76con2d 109 . 2  |-  ( A  e.  V  ->  ( ps  ->  -.  A. x  -.  ph ) )
8 df-ex 1548 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
97, 8syl6ibr 219 1  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1546   E.wex 1547   F/wnf 1550    = wceq 1649    e. wcel 1717   F/_wnfc 2511
This theorem is referenced by:  spcegv  2981  rspce  2991  euotd  4399  rspcegf  27363  stoweidlem36  27454  stoweidlem46  27464  bnj607  28626  bnj1491  28765
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902
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