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Theorem spesbc 3387
Description: Existence form of spsbc 3318. (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 3315 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 rspesbca 3386 . . 3  |-  ( ( A  e.  _V  /\  [. A  /  x ]. ph )  ->  E. x  e.  _V  ph )
31, 2mpancom 673 . 2  |-  ( [. A  /  x ]. ph  ->  E. x  e.  _V  ph )
4 rexv 3102 . 2  |-  ( E. x  e.  _V  ph  <->  E. x ph )
53, 4sylib 199 1  |-  ( [. A  /  x ]. ph  ->  E. x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1659    e. wcel 1870   E.wrex 2783   _Vcvv 3087   [.wsbc 3305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-v 3089  df-sbc 3306
This theorem is referenced by:  spesbcd  3388  opelopabsb  4731  sbccomieg  35345  frege124d  35992  sbiota1  36422
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