ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.33bdc Unicode version

Theorem 19.33bdc 1566
Description: Converse of 19.33 1418 given  -.  ( E. x ph  /\ 
E. x ps ) and a decidability condition. Compare Theorem 19.33 of [Margaris] p. 90. For a version which does not require a decidability condition, see 19.33b2 1565 (Contributed by Jim Kingdon, 23-Apr-2018.)
Assertion
Ref Expression
19.33bdc  |-  (DECID  E. x ph  ->  ( -.  ( E. x ph  /\  E. x ps )  ->  ( A. x ( ph  \/  ps )  <->  ( A. x ph  \/  A. x ps ) ) ) )

Proof of Theorem 19.33bdc
StepHypRef Expression
1 ianordc 837 . 2  |-  (DECID  E. x ph  ->  ( -.  ( E. x ph  /\  E. x ps )  <->  ( -.  E. x ph  \/  -.  E. x ps ) ) )
2 19.33b2 1565 . 2  |-  ( ( -.  E. x ph  \/  -.  E. x ps )  ->  ( A. x ( ph  \/  ps )  <->  ( A. x ph  \/  A. x ps ) ) )
31, 2syl6bi 161 1  |-  (DECID  E. x ph  ->  ( -.  ( E. x ph  /\  E. x ps )  ->  ( A. x ( ph  \/  ps )  <->  ( A. x ph  \/  A. x ps ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780   A.wal 1287   E.wex 1426
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-gen 1383  ax-ie2 1428
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-fal 1295
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator