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Theorem rspct 3013
Description: A closed version of rspc 3014. (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 2679 . . . 4  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
2 eleq1 2472 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
32adantr 452 . . . . . . . . 9  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( x  e.  B  <->  A  e.  B ) )
4 simpr 448 . . . . . . . . 9  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( ph  <->  ps ) )
53, 4imbi12d 312 . . . . . . . 8  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( ( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) )
65ex 424 . . . . . . 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 1565 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( x  =  A  ->  ( ( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps ) ) ) )
9 nfv 1626 . . . . . . 7  |-  F/ x  A  e.  B
10 rspct.1 . . . . . . 7  |-  F/ x ps
119, 10nfim 1828 . . . . . 6  |-  F/ x
( A  e.  B  ->  ps )
12 nfcv 2548 . . . . . 6  |-  F/_ x A
1311, 12spcgft 2996 . . . . 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 222 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ( A  e.  B  ->  ps ) ) ) )
1615com34 79 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A  e.  B  -> 
( A. x  e.  B  ph  ->  ps ) ) ) )
1716pm2.43d 46 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    <-> wb 177    /\ wa 359   A.wal 1546   F/wnf 1550    = wceq 1649    e. wcel 1721   A.wral 2674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-v 2926
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