Theorem pm5.6dc 926

Description: Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 927). (Contributed by Jim Kingdon, 2-Apr-2018.)

Assertion

Ref	Expression
pm5.6dc	⊢ (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒))))

Proof of Theorem pm5.6dc

Step	Hyp	Ref	Expression
1		impexp 263	. 2 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓 → 𝜒)))
2		dfordc 892	. . 3 ⊢ (DECID 𝜓 → ((𝜓 ∨ 𝜒) ↔ (¬ 𝜓 → 𝜒)))
3	2	imbi2d 230	. 2 ⊢ (DECID 𝜓 → ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → (¬ 𝜓 → 𝜒))))
4	1, 3	bitr4id 199	1 ⊢ (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834

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-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-dc 835

This theorem is referenced by: (None)