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Theorem rspce 2668
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 2194 . . . 4  |-  F/_ x A
2 nfv 1437 . . . . 5  |-  F/ x  A  e.  B
3 rspc.1 . . . . 5  |-  F/ x ps
42, 3nfan 1473 . . . 4  |-  F/ x
( A  e.  B  /\  ps )
5 eleq1 2116 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
6 rspc.2 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6anbi12d 450 . . . 4  |-  ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ( A  e.  B  /\  ps )
) )
81, 4, 7spcegf 2653 . . 3  |-  ( A  e.  B  ->  (
( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) ) )
98anabsi5 521 . 2  |-  ( ( A  e.  B  /\  ps )  ->  E. x
( x  e.  B  /\  ph ) )
10 df-rex 2329 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
119, 10sylibr 141 1  |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   F/wnf 1365   E.wex 1397    e. wcel 1409   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576
This theorem is referenced by:  rspcev  2673
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