ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.14dc Unicode version

Theorem pm5.14dc 881
Description: A decidable proposition is implied by or implies other propositions. Based on theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
Assertion
Ref Expression
pm5.14dc  |-  (DECID  ps  ->  ( ( ph  ->  ps )  \/  ( ps  ->  ch ) ) )

Proof of Theorem pm5.14dc
StepHypRef Expression
1 df-dc 805 . 2  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
2 ax-1 6 . . 3  |-  ( ps 
->  ( ph  ->  ps ) )
3 ax-in2 589 . . 3  |-  ( -. 
ps  ->  ( ps  ->  ch ) )
42, 3orim12i 733 . 2  |-  ( ( ps  \/  -.  ps )  ->  ( ( ph  ->  ps )  \/  ( ps  ->  ch ) ) )
51, 4sylbi 120 1  |-  (DECID  ps  ->  ( ( ph  ->  ps )  \/  ( ps  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 682  DECID wdc 804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-dc 805
This theorem is referenced by:  pm5.13dc  882
  Copyright terms: Public domain W3C validator