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Theorem spesbc 3085
Description: Existence form of spsbc 3016. (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 3013 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 rspesbca 3084 . . 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 2815 . 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 1531    e. wcel 1696   E.wrex 2557   _Vcvv 2801   [.wsbc 3004
This theorem is referenced by:  spesbcd  3086  opelopabsb  4291  sbccomieg  26973  sbiota1  27737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005
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