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Theorem 19.33bdc 1562
Description: Converse of 19.33 1414 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 1561 (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 833 . 2  |-  (DECID  E. x ph  ->  ( -.  ( E. x ph  /\  E. x ps )  <->  ( -.  E. x ph  \/  -.  E. x ps ) ) )
2 19.33b2 1561 . 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 662  DECID wdc 776   A.wal 1283   E.wex 1422
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-gen 1379  ax-ie2 1424
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-fal 1291
This theorem is referenced by: (None)
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