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Theorem spesbc 3072
Description: Existence form of spsbc 3003. (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 3000 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 rspesbca 3071 . . 3  |-  ( ( A  e.  _V  /\  [. A  /  x ]. ph )  ->  E. x  e.  _V  ph )
31, 2mpancom 650 . 2  |-  ( [. A  /  x ]. ph  ->  E. x  e.  _V  ph )
4 rexv 2802 . 2  |-  ( E. x  e.  _V  ph  <->  E. x ph )
53, 4sylib 188 1  |-  ( [. A  /  x ]. ph  ->  E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1528    e. wcel 1684   E.wrex 2544   _Vcvv 2788   [.wsbc 2991
This theorem is referenced by:  spesbcd  3073  opelopabsb  4275  sbccomieg  26870  sbiota1  27634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992
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