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Theorem rspce 3214
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 2629 . . . 4  |-  F/_ x A
2 nfv 1683 . . . . 5  |-  F/ x  A  e.  B
3 rspc.1 . . . . 5  |-  F/ x ps
42, 3nfan 1875 . . . 4  |-  F/ x
( A  e.  B  /\  ps )
5 eleq1 2539 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
6 rspc.2 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6anbi12d 710 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
81, 4, 7spcegf 3199 . . 3  |-  ( A  e.  B  ->  (
( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) ) )
98anabsi5 815 . 2  |-  ( ( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) )
10 df-rex 2823 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
119, 10sylibr 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   E.wrex 2818
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 2823  df-v 3120
This theorem is referenced by:  rspcev  3219  ac6c4  8873  fsumcom2  13569  infcvgaux1i  13648  iunmbl2  21835  esumcvg  27917  fprodcom2  29041  sdclem1  30163
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