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Theorem rspct 2715
Description: A closed version of rspc 2716. (Contributed by Andrew Salmon, 6-Jun-2011.)
Hypothesis
Ref Expression
rspct.1  |-  F/ x ps
Assertion
Ref Expression
rspct  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ps ) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem rspct
StepHypRef Expression
1 df-ral 2364 . . . 4  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
2 eleq1 2150 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
32adantr 270 . . . . . . . . 9  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( x  e.  B  <->  A  e.  B ) )
4 simpr 108 . . . . . . . . 9  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( ph  <->  ps ) )
53, 4imbi12d 232 . . . . . . . 8  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( ( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) )
65ex 113 . . . . . . 7  |-  ( x  =  A  ->  (
( ph  <->  ps )  ->  (
( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) ) )
76a2i 11 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps ) ) ) )
87alimi 1389 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( x  =  A  ->  ( ( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps ) ) ) )
9 nfv 1466 . . . . . . 7  |-  F/ x  A  e.  B
10 rspct.1 . . . . . . 7  |-  F/ x ps
119, 10nfim 1509 . . . . . 6  |-  F/ x
( A  e.  B  ->  ps )
12 nfcv 2228 . . . . . 6  |-  F/_ x A
1311, 12spcgft 2696 . . . . 5  |-  ( A. x ( x  =  A  ->  ( (
x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) )  ->  ( A  e.  B  ->  ( A. x ( x  e.  B  ->  ph )  ->  ( A  e.  B  ->  ps ) ) ) )
148, 13syl 14 . . . 4  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x ( x  e.  B  ->  ph )  ->  ( A  e.  B  ->  ps ) ) ) )
151, 14syl7bi 163 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ( A  e.  B  ->  ps ) ) ) )
1615com34 82 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A  e.  B  -> 
( A. x  e.  B  ph  ->  ps ) ) ) )
1716pm2.43d 49 1  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289   F/wnf 1394    e. wcel 1438   A.wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621
This theorem is referenced by:  sumdc2  11356
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