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Theorem spcgft 3020
Description: A closed version of spcgf 3023. (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 179 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
21imim2i 14 . . 3  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ph  ->  ps ) ) )
32alimi 1568 . 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 3019 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  B  ->  ( A. x ph  ->  ps ) ) )
73, 6syl 16 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 177   A.wal 1549   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558
This theorem is referenced by:  spcgf  3023  rspct  3037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950
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