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Theorem spcimegft 2677
Description: A closed version of spcimegf 2680. (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 2611 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 spcimgft.2 . . . . 5  |-  F/_ x A
32issetf 2607 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
4 exim 1531 . . . 4  |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) )  -> 
( E. x  x  =  A  ->  E. x
( ps  ->  ph )
) )
53, 4syl5bi 150 . . 3  |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) )  -> 
( A  e.  _V  ->  E. x ( ps 
->  ph ) ) )
6 spcimgft.1 . . . 4  |-  F/ x ps
7619.37-1 1605 . . 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 1283    = wceq 1285   F/wnf 1390   E.wex 1422    e. wcel 1434   F/_wnfc 2207   _Vcvv 2602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604
This theorem is referenced by:  spcegft  2678  spcimegf  2680  spcimedv  2685
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