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Theorem rspce 2836
Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
Hypotheses
Ref Expression
rspc.1  |-  F/ x ps
rspc.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rspce  |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem rspce
StepHypRef Expression
1 nfcv 2319 . . . 4  |-  F/_ x A
2 nfv 1528 . . . . 5  |-  F/ x  A  e.  B
3 rspc.1 . . . . 5  |-  F/ x ps
42, 3nfan 1565 . . . 4  |-  F/ x
( A  e.  B  /\  ps )
5 eleq1 2240 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
6 rspc.2 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6anbi12d 473 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
81, 4, 7spcegf 2820 . . 3  |-  ( A  e.  B  ->  (
( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) ) )
98anabsi5 579 . 2  |-  ( ( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) )
10 df-rex 2461 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
119, 10sylibr 134 1  |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   F/wnf 1460   E.wex 1492    e. wcel 2148   E.wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739
This theorem is referenced by:  rspcev  2841  bezoutlemmain  11993
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