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

Theorem orimdidc 850
Description: Disjunction distributes over implication. The forward direction, pm2.76 757, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 849. (Contributed by Jim Kingdon, 1-Apr-2018.)
Assertion
Ref Expression
orimdidc (DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒))))

Proof of Theorem orimdidc
StepHypRef Expression
1 pm2.76 757 . 2 ((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
2 pm2.85dc 849 . 2 (DECID 𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))
31, 2impbid2 141 1 (DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒))))
Colors of variables: wff set class
Syntax hints:  wi 4  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-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by:  orbididc  899
  Copyright terms: Public domain W3C validator