Theorem pm5.62dc 940

Description: Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)

Assertion

Ref	Expression
pm5.62dc	⊢ (DECID 𝜓 → (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))

Proof of Theorem pm5.62dc

Step	Hyp	Ref	Expression
1		df-dc 830	. 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
2		ordir 812	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ ((𝜑 ∨ ¬ 𝜓) ∧ (𝜓 ∨ ¬ 𝜓)))
3	2	simplbi 272	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) → (𝜑 ∨ ¬ 𝜓))
4	2	simplbi2 383	. . . 4 ⊢ ((𝜑 ∨ ¬ 𝜓) → ((𝜓 ∨ ¬ 𝜓) → ((𝜑 ∧ 𝜓) ∨ ¬ 𝜓)))
5	4	com12 30	. . 3 ⊢ ((𝜓 ∨ ¬ 𝜓) → ((𝜑 ∨ ¬ 𝜓) → ((𝜑 ∧ 𝜓) ∨ ¬ 𝜓)))
6	3, 5	impbid2 142	. 2 ⊢ ((𝜓 ∨ ¬ 𝜓) → (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
7	1, 6	sylbi 120	1 ⊢ (DECID 𝜓 → (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ 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-io 704

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

This theorem is referenced by: (None)