Theorem orimdidc 907

Description: Disjunction distributes over implication. The forward direction, pm2.76 809, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 906. (Contributed by Jim Kingdon, 1-Apr-2018.)

Assertion

Ref	Expression
orimdidc	⊢ (DECID 𝜑 → ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒))))

Proof of Theorem orimdidc

Step	Hyp	Ref	Expression
1		pm2.76 809	. 2 ⊢ ((𝜑 ∨ (𝜓 → 𝜒)) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))
2		pm2.85dc 906	. 2 ⊢ (DECID 𝜑 → (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒))))
3	1, 2	impbid2 143	1 ⊢ (DECID 𝜑 → ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒))))

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 709  DECID wdc 835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-dc 836

This theorem is referenced by:  orbididc  955