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Theorem rspcegf 31004
Description: A version of rspcev 3214 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 2642 . . . . 5  |-  F/ x  A  e.  B
4 rspcegf.1 . . . . 5  |-  F/ x ps
53, 4nfan 1875 . . . 4  |-  F/ x
( A  e.  B  /\  ps )
6 eleq1 2539 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
7 rspcegf.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
86, 7anbi12d 710 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
91, 5, 8spcegf 3194 . . 3  |-  ( A  e.  B  ->  (
( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) ) )
109anabsi5 815 . 2  |-  ( ( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) )
11 df-rex 2820 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
1210, 11sylibr 212 1  |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596   F/wnf 1599    e. wcel 1767   F/_wnfc 2615   E.wrex 2815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-v 3115
This theorem is referenced by:  stoweidlem46  31374
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