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Theorem spesbc 3048
Description: Existence form of spsbc 2974. (Contributed by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
spesbc  |-  ( [. A  /  x ]. ph  ->  E. x ph )

Proof of Theorem spesbc
StepHypRef Expression
1 sbcex 2971 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 rspesbca 3047 . . 3  |-  ( ( A  e.  _V  /\  [. A  /  x ]. ph )  ->  E. x  e.  _V  ph )
31, 2mpancom 422 . 2  |-  ( [. A  /  x ]. ph  ->  E. x  e.  _V  ph )
4 rexv 2755 . 2  |-  ( E. x  e.  _V  ph  <->  E. x ph )
53, 4sylib 122 1  |-  ( [. A  /  x ]. ph  ->  E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1492    e. wcel 2148   E.wrex 2456   _Vcvv 2737   [.wsbc 2962
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  df-sbc 2963
This theorem is referenced by:  spesbcd  3049  opelopabsb  4260
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