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Theorem spcimegft 2738
Description: A closed version of spcimegf 2741. (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 2671 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 spcimgft.2 . . . . 5  |-  F/_ x A
32issetf 2667 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
4 exim 1563 . . . 4  |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) )  -> 
( E. x  x  =  A  ->  E. x
( ps  ->  ph )
) )
53, 4syl5bi 151 . . 3  |-  ( A. x ( x  =  A  ->  ( ps  ->  ph ) )  -> 
( A  e.  _V  ->  E. x ( ps 
->  ph ) ) )
6 spcimgft.1 . . . 4  |-  F/ x ps
7619.37-1 1637 . . 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 1314    = wceq 1316   F/wnf 1421   E.wex 1453    e. wcel 1465   F/_wnfc 2245   _Vcvv 2660
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  spcegft  2739  spcimegf  2741  spcimedv  2746
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