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Theorem rspct 3118
Description: A closed version of rspc 3119. (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 2719 . . . 4  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
2 eleq1 2494 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
32adantr 466 . . . . . . . . 9  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( x  e.  B  <->  A  e.  B ) )
4 simpr 462 . . . . . . . . 9  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( ph  <->  ps ) )
53, 4imbi12d 321 . . . . . . . 8  |-  ( ( x  =  A  /\  ( ph  <->  ps ) )  -> 
( ( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) )
65ex 435 . . . . . . 7  |-  ( x  =  A  ->  (
( ph  <->  ps )  ->  (
( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) ) )
76a2i 14 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps ) ) ) )
87alimi 1678 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( x  =  A  ->  ( ( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps ) ) ) )
9 nfv 1755 . . . . . . 7  |-  F/ x  A  e.  B
10 rspct.1 . . . . . . 7  |-  F/ x ps
119, 10nfim 1980 . . . . . 6  |-  F/ x
( A  e.  B  ->  ps )
12 nfcv 2569 . . . . . 6  |-  F/_ x A
1311, 12spcgft 3101 . . . . 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 17 . . . 4  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x ( x  e.  B  ->  ph )  ->  ( A  e.  B  ->  ps ) ) ) )
151, 14syl7bi 233 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ( A  e.  B  ->  ps ) ) ) )
1615com34 86 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  B  ->  ( A  e.  B  -> 
( A. x  e.  B  ph  ->  ps ) ) ) )
1716pm2.43d 50 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 187    /\ wa 370   A.wal 1435    = wceq 1437   F/wnf 1661    e. wcel 1872   A.wral 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-v 3024
This theorem is referenced by: (None)
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