ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.62dc Unicode version

Theorem pm5.62dc 891
Description: Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
Assertion
Ref Expression
pm5.62dc  |-  (DECID  ps  ->  ( ( ( ph  /\  ps )  \/  -.  ps )  <->  ( ph  \/  -.  ps ) ) )

Proof of Theorem pm5.62dc
StepHypRef Expression
1 df-dc 781 . 2  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
2 ordir 766 . . . 4  |-  ( ( ( ph  /\  ps )  \/  -.  ps )  <->  ( ( ph  \/  -.  ps )  /\  ( ps  \/  -.  ps )
) )
32simplbi 268 . . 3  |-  ( ( ( ph  /\  ps )  \/  -.  ps )  ->  ( ph  \/  -.  ps ) )
42simplbi2 377 . . . 4  |-  ( (
ph  \/  -.  ps )  ->  ( ( ps  \/  -.  ps )  ->  (
( ph  /\  ps )  \/  -.  ps ) ) )
54com12 30 . . 3  |-  ( ( ps  \/  -.  ps )  ->  ( ( ph  \/  -.  ps )  -> 
( ( ph  /\  ps )  \/  -.  ps ) ) )
63, 5impbid2 141 . 2  |-  ( ( ps  \/  -.  ps )  ->  ( ( (
ph  /\  ps )  \/  -.  ps )  <->  ( ph  \/  -.  ps ) ) )
71, 6sylbi 119 1  |-  (DECID  ps  ->  ( ( ( ph  /\  ps )  \/  -.  ps )  <->  ( ph  \/  -.  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780
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
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator