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Theorem spcimgft 2828
Description: A closed version of spcimgf 2832. (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 2763 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 spcimgft.2 . . . . 5  |-  F/_ x A
32issetf 2759 . . . 4  |-  ( A  e.  _V  <->  E. x  x  =  A )
4 exim 1610 . . . 4  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( E. x  x  =  A  ->  E. x
( ph  ->  ps )
) )
53, 4biimtrid 152 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  _V  ->  E. x ( ph  ->  ps ) ) )
6 spcimgft.1 . . . 4  |-  F/ x ps
7619.36-1 1684 . . 3  |-  ( E. x ( ph  ->  ps )  ->  ( A. x ph  ->  ps )
)
85, 7syl6 33 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  _V  ->  ( A. x ph  ->  ps ) ) )
91, 8syl5 32 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 1362    = wceq 1364   F/wnf 1471   E.wex 1503    e. wcel 2160   F/_wnfc 2319   _Vcvv 2752
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754
This theorem is referenced by:  spcgft  2829  spcimgf  2832  spcimdv  2836
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