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Theorem rspce 2955
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 2494 . . . 4  |-  F/_ x A
2 nfv 1619 . . . . 5  |-  F/ x  A  e.  B
3 rspc.1 . . . . 5  |-  F/ x ps
42, 3nfan 1829 . . . 4  |-  F/ x
( A  e.  B  /\  ps )
5 eleq1 2418 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
6 rspc.2 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6anbi12d 691 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
81, 4, 7spcegf 2940 . . 3  |-  ( A  e.  B  ->  (
( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) ) )
98anabsi5 790 . 2  |-  ( ( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) )
10 df-rex 2625 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
119, 10sylibr 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 1541   F/wnf 1544    = wceq 1642    e. wcel 1710   E.wrex 2620
This theorem is referenced by:  rspcev  2960  ac6c4  8195  fsumcom2  12328  infcvgaux1i  12406  iunmbl2  19012  esumcst  23721  esumcvg  23742  ballotlemic  24013  ballotlem1c  24014  sdclem1  25777  stoweidlem14  27086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rex 2625  df-v 2866
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