Theorem pm5.71 1025

Description: Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.)

Assertion

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

Proof of Theorem pm5.71

Step	Hyp	Ref	Expression
1		orel2 888	. . . 4 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) → 𝜑))
2		orc 864	. . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	1, 2	impbid1 224	. . 3 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑))
4	3	anbi1d 630	. 2 ⊢ (¬ 𝜓 → (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜒)))
5		pm2.21 123	. . 3 ⊢ (¬ 𝜒 → (𝜒 → ((𝜑 ∨ 𝜓) ↔ 𝜑)))
6	5	pm5.32rd 578	. 2 ⊢ (¬ 𝜒 → (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜒)))
7	4, 6	ja 186	1 ⊢ ((𝜓 → ¬ 𝜒) → (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845

This theorem is referenced by: (None)