Theorem pm5.7dc 954

Description: Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 953. (Contributed by Jim Kingdon, 2-Apr-2018.)

Assertion

Ref	Expression
pm5.7dc	⊢ (DECID 𝜒 → (((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)) ↔ (𝜒 ∨ (𝜑 ↔ 𝜓))))

Proof of Theorem pm5.7dc

Step	Hyp	Ref	Expression
1		orbididc 953	. 2 ⊢ (DECID 𝜒 → ((𝜒 ∨ (𝜑 ↔ 𝜓)) ↔ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))))
2		orcom 728	. . 3 ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜑 ∨ 𝜒))
3		orcom 728	. . 3 ⊢ ((𝜒 ∨ 𝜓) ↔ (𝜓 ∨ 𝜒))
4	2, 3	bibi12i 229	. 2 ⊢ (((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)) ↔ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)))
5	1, 4	bitr2di 197	1 ⊢ (DECID 𝜒 → (((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)) ↔ (𝜒 ∨ (𝜑 ↔ 𝜓))))

Syntax hints:   → wi 4   ↔ 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-in2 615  ax-io 709

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

This theorem is referenced by: (None)