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Theorem spcimegf 2818
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.2 . . 3  |-  F/ x ps
2 spcimgf.1 . . 3  |-  F/_ x A
31, 2spcimegft 2815 . 2  |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) )  -> 
( A  e.  V  ->  ( ps  ->  E. x ph ) ) )
4 spcimegf.3 . 2  |-  ( x  =  A  ->  ( ps  ->  ph ) )
53, 4mpg 1451 1  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   F/wnf 1460   E.wex 1492    e. wcel 2148   F/_wnfc 2306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by: (None)
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