Theorem pm5.71dc 868

Description: Decidable proposition version of theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.) (Modified for decidability by Jim Kingdon, 19-Apr-2018.)

Assertion

Ref	Expression
pm5.71dc	⊢ (DECID 𝜓 → ((𝜓 → ¬ 𝜒) → (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜒))))

Proof of Theorem pm5.71dc

Step	Hyp	Ref	Expression
1		orel2 645	. . . . 5 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) → 𝜑))
2		orc 633	. . . . 5 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	1, 2	impbid1 130	. . . 4 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑))
4	3	anbi1d 438	. . 3 ⊢ (¬ 𝜓 → (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜒)))
5	4	a1i 9	. 2 ⊢ (DECID 𝜓 → (¬ 𝜓 → (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜒))))
6		pm2.21 547	. . 3 ⊢ (¬ 𝜒 → (𝜒 → ((𝜑 ∨ 𝜓) ↔ 𝜑)))
7	6	pm5.32rd 424	. 2 ⊢ (¬ 𝜒 → (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜒)))
8	5, 7	jadc 760	1 ⊢ (DECID 𝜓 → ((𝜓 → ¬ 𝜒) → (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜒))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by: (None)