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Theorem spcimgft 3029
Description: A closed version of spcimgf 3031. (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 2966 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 spcimgft.2 . . . . 5  |-  F/_ x A
32issetf 2963 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
4 exim 1585 . . . 4  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( E. x  x  =  A  ->  E. x
( ph  ->  ps )
) )
53, 4syl5bi 210 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  _V  ->  E. x ( ph  ->  ps ) ) )
6 spcimgft.1 . . . 4  |-  F/ x ps
7619.36 1893 . . 3  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  ps ) )
85, 7syl6ib 219 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  _V  ->  ( A. x ph  ->  ps ) ) )
91, 8syl5 31 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 1550   E.wex 1551   F/wnf 1554    = wceq 1653    e. wcel 1726   F/_wnfc 2561   _Vcvv 2958
This theorem is referenced by:  spcgft  3030  spcimgf  3031  spcimdv  3035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960
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