ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  spesbc Unicode version

Theorem spesbc 2924
Description: Existence form of spsbc 2851. (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 2848 . . 3  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 rspesbca 2923 . . 3  |-  ( ( A  e.  _V  /\  [. A  /  x ]. ph )  ->  E. x  e.  _V  ph )
31, 2mpancom 413 . 2  |-  ( [. A  /  x ]. ph  ->  E. x  e.  _V  ph )
4 rexv 2637 . 2  |-  ( E. x  e.  _V  ph  <->  E. x ph )
53, 4sylib 120 1  |-  ( [. A  /  x ]. ph  ->  E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1426    e. wcel 1438   E.wrex 2360   _Vcvv 2619   [.wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2841
This theorem is referenced by:  spesbcd  2925  opelopabsb  4087
  Copyright terms: Public domain W3C validator