Theorem pm5.6dc 911

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 912). (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		dfordc 877	. . 3 ⊢ (DECID 𝜓 → ((𝜓 ∨ 𝜒) ↔ (¬ 𝜓 → 𝜒)))
2	1	imbi2d 229	. 2 ⊢ (DECID 𝜓 → ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → (¬ 𝜓 → 𝜒))))
3		impexp 261	. 2 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓 → 𝜒)))
4	2, 3	syl6rbbr 198	1 ⊢ (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  DECID wdc 819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-dc 820

This theorem is referenced by: (None)