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Theorem rspct 3061
Description: A closed version of rspc 3062. (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 2715 . . . 4  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
2 eleq1 2498 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
32adantr 465 . . . . . . . . 9  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( x  e.  B  <->  A  e.  B ) )
4 simpr 461 . . . . . . . . 9  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( ph  <->  ps ) )
53, 4imbi12d 320 . . . . . . . 8  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( ( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) )
65ex 434 . . . . . . 7  |-  ( x  =  A  ->  (
( ph  <->  ps )  ->  (
( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) ) )
76a2i 13 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps ) ) ) )
87alimi 1604 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( x  =  A  ->  ( ( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps ) ) ) )
9 nfv 1673 . . . . . . 7  |-  F/ x  A  e.  B
10 rspct.1 . . . . . . 7  |-  F/ x ps
119, 10nfim 1852 . . . . . 6  |-  F/ x
( A  e.  B  ->  ps )
12 nfcv 2574 . . . . . 6  |-  F/_ x A
1311, 12spcgft 3044 . . . . 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 16 . . . 4  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x ( x  e.  B  ->  ph )  ->  ( A  e.  B  ->  ps ) ) ) )
151, 14syl7bi 230 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ( A  e.  B  ->  ps ) ) ) )
1615com34 83 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A  e.  B  -> 
( A. x  e.  B  ph  ->  ps ) ) ) )
1716pm2.43d 48 1  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369   F/wnf 1589    e. wcel 1756   A.wral 2710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-v 2969
This theorem is referenced by: (None)
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