 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  orimdidc GIF version

Theorem orimdidc 812
 Description: Disjunction distributes over implication. The forward direction, pm2.76 721, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 811. (Contributed by Jim Kingdon, 1-Apr-2018.)
Assertion
Ref Expression
orimdidc (DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒))))

Proof of Theorem orimdidc
StepHypRef Expression
1 pm2.76 721 . 2 ((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
2 pm2.85dc 811 . 2 (DECID 𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))
31, 2impbid2 131 1 (DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 629  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  orbididc  860
 Copyright terms: Public domain W3C validator