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Theorem orimdidc 846
 Description: Disjunction distributes over implication. The forward direction, pm2.76 755, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 845. (Contributed by Jim Kingdon, 1-Apr-2018.)
Assertion
Ref Expression
orimdidc (DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒))))

Proof of Theorem orimdidc
StepHypRef Expression
1 pm2.76 755 . 2 ((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
2 pm2.85dc 845 . 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 662  DECID wdc 776 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 578  ax-io 663 This theorem depends on definitions:  df-bi 115  df-dc 777 This theorem is referenced by:  orbididc  895
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