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Theorem spcimgft 2859
Description: A closed version of spcimgf 2861. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgft.1  |-  F/ x ps
spcimgft.2  |-  F/_ x A
Assertion
Ref Expression
spcimgft  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  B  ->  ( A. x ph  ->  ps ) ) )

Proof of Theorem spcimgft
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 spcimgft.2 . . . . 5  |-  F/_ x A
32issetf 2793 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
4 exim 1562 . . . 4  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( E. x  x  =  A  ->  E. x
( ph  ->  ps )
) )
53, 4syl5bi 208 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  _V  ->  E. x ( ph  ->  ps ) ) )
6 spcimgft.1 . . . 4  |-  F/ x ps
7619.36 1807 . . 3  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  ps ) )
85, 7syl6ib 217 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  _V  ->  ( A. x ph  ->  ps ) ) )
91, 8syl5 28 1  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  B  ->  ( A. x ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   E.wex 1528   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788
This theorem is referenced by:  spcgft  2860  spcimgf  2861  spcimdv  2865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
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