Theorem pm5.55dc 908

Description: A disjunction is equivalent to one of its disjuncts, given a decidable disjunct. Based on theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)

Assertion

Ref	Expression
pm5.55dc	⊢ (DECID 𝜑 → (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓)))

Proof of Theorem pm5.55dc

Step	Hyp	Ref	Expression
1		df-dc 830	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		biort 824	. . . 4 ⊢ (𝜑 → (𝜑 ↔ (𝜑 ∨ 𝜓)))
3	2	bicomd 140	. . 3 ⊢ (𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜑))
4		biorf 739	. . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))
5	4	bicomd 140	. . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓))
6	3, 5	orim12i 754	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓)))
7	1, 6	sylbi 120	1 ⊢ (DECID 𝜑 → (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 703  DECID wdc 829

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-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-dc 830

This theorem is referenced by: (None)