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Theorem spcimegf 2767
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 2764 . 2  |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) )  -> 
( A  e.  V  ->  ( ps  ->  E. x ph ) ) )
4 spcimegf.3 . 2  |-  ( x  =  A  ->  ( ps  ->  ph ) )
53, 4mpg 1427 1  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   F/wnf 1436   E.wex 1468    e. wcel 1480   F/_wnfc 2268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by: (None)
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