MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spesbc Structured version   Unicode version

Theorem spesbc 3242
Description: Existence form of spsbc 3173. (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 3170 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 rspesbca 3241 . . 3  |-  ( ( A  e.  _V  /\  [. A  /  x ]. ph )  ->  E. x  e.  _V  ph )
31, 2mpancom 651 . 2  |-  ( [. A  /  x ]. ph  ->  E. x  e.  _V  ph )
4 rexv 2970 . 2  |-  ( E. x  e.  _V  ph  <->  E. x ph )
53, 4sylib 189 1  |-  ( [. A  /  x ]. ph  ->  E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1550    e. wcel 1725   E.wrex 2706   _Vcvv 2956   [.wsbc 3161
This theorem is referenced by:  spesbcd  3243  opelopabsb  4465  sbccomieg  26849  sbiota1  27611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-sbc 3162
  Copyright terms: Public domain W3C validator