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Theorem rspcegf 27782
Description: A version of rspcev 3061 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
rspcegf.1  |-  F/ x ps
rspcegf.2  |-  F/_ x A
rspcegf.3  |-  F/_ x B
rspcegf.4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rspcegf  |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )

Proof of Theorem rspcegf
StepHypRef Expression
1 rspcegf.2 . . . 4  |-  F/_ x A
2 rspcegf.3 . . . . . 6  |-  F/_ x B
31, 2nfel 2587 . . . . 5  |-  F/ x  A  e.  B
4 rspcegf.1 . . . . 5  |-  F/ x ps
53, 4nfan 1849 . . . 4  |-  F/ x
( A  e.  B  /\  ps )
6 eleq1 2503 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
7 rspcegf.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
86, 7anbi12d 693 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
91, 5, 8spcegf 3041 . . 3  |-  ( A  e.  B  ->  (
( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) ) )
109anabsi5 792 . 2  |-  ( ( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) )
11 df-rex 2718 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
1210, 11sylibr 205 1  |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551   F/wnf 1554    = wceq 1654    e. wcel 1728   F/_wnfc 2566   E.wrex 2713
This theorem is referenced by:  stoweidlem46  27883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2718  df-v 2967
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