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Theorem spesbc 3429
Description: Existence form of spsbc 3349. (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 3346 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 rspesbca 3428 . . 3  |-  ( ( A  e.  _V  /\  [. A  /  x ]. ph )  ->  E. x  e.  _V  ph )
31, 2mpancom 669 . 2  |-  ( [. A  /  x ]. ph  ->  E. x  e.  _V  ph )
4 rexv 3133 . 2  |-  ( E. x  e.  _V  ph  <->  E. x ph )
53, 4sylib 196 1  |-  ( [. A  /  x ]. ph  ->  E. x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1596    e. wcel 1767   E.wrex 2818   _Vcvv 3118   [.wsbc 3336
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-or 370  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-ral 2822  df-rex 2823  df-v 3120  df-sbc 3337
This theorem is referenced by:  spesbcd  3430  opelopabsb  4763  sbccomieg  30645  sbiota1  31234
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