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Theorem spesbc 3387
Description: Existence form of spsbc 3307. (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 3304 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 rspesbca 3386 . . 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 3093 . 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 1587    e. wcel 1758   E.wrex 2800   _Vcvv 3078   [.wsbc 3294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-sbc 3295
This theorem is referenced by:  spesbcd  3388  opelopabsb  4708  sbccomieg  29280  sbiota1  29837
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