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Theorem spcimegft 2815
Description: A closed version of spcimegf 2818. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgft.1  |-  F/ x ps
spcimgft.2  |-  F/_ x A
Assertion
Ref Expression
spcimegft  |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) )  -> 
( A  e.  B  ->  ( ps  ->  E. x ph ) ) )

Proof of Theorem spcimegft
StepHypRef Expression
1 elex 2748 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 spcimgft.2 . . . . 5  |-  F/_ x A
32issetf 2744 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
4 exim 1599 . . . 4  |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) )  -> 
( E. x  x  =  A  ->  E. x
( ps  ->  ph )
) )
53, 4biimtrid 152 . . 3  |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) )  -> 
( A  e.  _V  ->  E. x ( ps 
->  ph ) ) )
6 spcimgft.1 . . . 4  |-  F/ x ps
7619.37-1 1674 . . 3  |-  ( E. x ( ps  ->  ph )  ->  ( ps  ->  E. x ph )
)
85, 7syl6 33 . 2  |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) )  -> 
( A  e.  _V  ->  ( ps  ->  E. x ph ) ) )
91, 8syl5 32 1  |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) )  -> 
( A  e.  B  ->  ( ps  ->  E. x ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1351    = wceq 1353   F/wnf 1460   E.wex 1492    e. wcel 2148   F/_wnfc 2306   _Vcvv 2737
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:  spcegft  2816  spcimegf  2818  spcimedv  2823
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