ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.6r Unicode version

Theorem pm5.6r 847
Description: Conjunction in antecedent versus disjunction in consequent. One direction of Theorem *5.6 of [WhiteheadRussell] p. 125. If  ps is decidable, the converse also holds (see pm5.6dc 846). (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
pm5.6r  |-  ( (
ph  ->  ( ps  \/  ch ) )  ->  (
( ph  /\  -.  ps )  ->  ch ) )

Proof of Theorem pm5.6r
StepHypRef Expression
1 pm2.53 651 . . 3  |-  ( ( ps  \/  ch )  ->  ( -.  ps  ->  ch ) )
21imim2i 12 . 2  |-  ( (
ph  ->  ( ps  \/  ch ) )  ->  ( ph  ->  ( -.  ps  ->  ch ) ) )
32impd 246 1  |-  ( (
ph  ->  ( ps  \/  ch ) )  ->  (
( ph  /\  -.  ps )  ->  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    \/ wo 639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  ssundifim  3334
  Copyright terms: Public domain W3C validator