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Theorem spcimegf 2862
Description: Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgf.1  |-  F/_ x A
spcimgf.2  |-  F/ x ps
spcimegf.3  |-  ( x  =  A  ->  ( ps  ->  ph ) )
Assertion
Ref Expression
spcimegf  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)

Proof of Theorem spcimegf
StepHypRef Expression
1 spcimgf.1 . . . 4  |-  F/_ x A
2 spcimgf.2 . . . . 5  |-  F/ x ps
32nfn 1765 . . . 4  |-  F/ x  -.  ps
4 spcimegf.3 . . . . 5  |-  ( x  =  A  ->  ( ps  ->  ph ) )
54con3d 125 . . . 4  |-  ( x  =  A  ->  ( -.  ph  ->  -.  ps )
)
61, 3, 5spcimgf 2861 . . 3  |-  ( A  e.  V  ->  ( A. x  -.  ph  ->  -. 
ps ) )
76con2d 107 . 2  |-  ( A  e.  V  ->  ( ps  ->  -.  A. x  -.  ph ) )
8 df-ex 1529 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
97, 8syl6ibr 218 1  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527   E.wex 1528   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
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