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Theorem rspcegf 27797
Description: A version of rspcev 2897 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 2440 . . . . 5  |-  F/ x  A  e.  B
4 rspcegf.1 . . . . 5  |-  F/ x ps
53, 4nfan 1783 . . . 4  |-  F/ x
( A  e.  B  /\  ps )
6 eleq1 2356 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
7 rspcegf.4 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
86, 7anbi12d 691 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
91, 5, 8spcegf 2877 . . 3  |-  ( A  e.  B  ->  (
( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) ) )
109anabsi5 790 . 2  |-  ( ( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) )
11 df-rex 2562 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
1210, 11sylibr 203 1  |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419   E.wrex 2557
This theorem is referenced by:  stoweidlem46  27898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803
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