Theorem pm5.63dc 887

Description: Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)

Assertion

Ref	Expression
pm5.63dc	⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ (¬ 𝜑 ∧ 𝜓))))

Proof of Theorem pm5.63dc

Step	Hyp	Ref	Expression
1		df-dc 776	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		ordi 762	. . . 4 ⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜓)) ↔ ((𝜑 ∨ ¬ 𝜑) ∧ (𝜑 ∨ 𝜓)))
3	2	simplbi2 377	. . 3 ⊢ ((𝜑 ∨ ¬ 𝜑) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ (¬ 𝜑 ∧ 𝜓))))
4	1, 3	sylbi 119	. 2 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) → (𝜑 ∨ (¬ 𝜑 ∧ 𝜓))))
5	2	simprbi 269	. 2 ⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜓)) → (𝜑 ∨ 𝜓))
6	4, 5	impbid1 140	1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ (¬ 𝜑 ∧ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by: (None)