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Theorem spcgft 2873
Description: A closed version of spcgf 2876. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgft.1  |-  F/ x ps
spcimgft.2  |-  F/_ x A
Assertion
Ref Expression
spcgft  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x ph  ->  ps ) ) )

Proof of Theorem spcgft
StepHypRef Expression
1 bi1 178 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
21imim2i 13 . . 3  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ph  ->  ps ) ) )
32alimi 1549 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( x  =  A  ->  ( ph  ->  ps ) ) )
4 spcimgft.1 . . 3  |-  F/ x ps
5 spcimgft.2 . . 3  |-  F/_ x A
64, 5spcimgft 2872 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  B  ->  ( A. x ph  ->  ps ) ) )
73, 6syl 15 1  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419
This theorem is referenced by:  spcgf  2876  rspct  2890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803
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