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Theorem spcgft 2647
Description: A closed version of spcgf 2652. (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 115 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
21imim2i 12 . . 3  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ph  ->  ps ) ) )
32alimi 1360 . 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 2646 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  B  ->  ( A. x ph  ->  ps ) ) )
73, 6syl 14 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 102   A.wal 1257    = wceq 1259   F/wnf 1365    e. wcel 1409   F/_wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  spcgf  2652  rspct  2666
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