Theorem pm5.53 715

Description: Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm5.53	⊢ ((((𝜑 ∨ 𝜓) ∨ 𝜒) → 𝜃) ↔ (((𝜑 → 𝜃) ∧ (𝜓 → 𝜃)) ∧ (𝜒 → 𝜃)))

Proof of Theorem pm5.53

Step	Hyp	Ref	Expression
1		jaob 631	. 2 ⊢ ((((𝜑 ∨ 𝜓) ∨ 𝜒) → 𝜃) ↔ (((𝜑 ∨ 𝜓) → 𝜃) ∧ (𝜒 → 𝜃)))
2		jaob 631	. . 3 ⊢ (((𝜑 ∨ 𝜓) → 𝜃) ↔ ((𝜑 → 𝜃) ∧ (𝜓 → 𝜃)))
3	2	anbi1i 431	. 2 ⊢ ((((𝜑 ∨ 𝜓) → 𝜃) ∧ (𝜒 → 𝜃)) ↔ (((𝜑 → 𝜃) ∧ (𝜓 → 𝜃)) ∧ (𝜒 → 𝜃)))
4	1, 3	bitri 173	1 ⊢ ((((𝜑 ∨ 𝜓) ∨ 𝜒) → 𝜃) ↔ (((𝜑 → 𝜃) ∧ (𝜓 → 𝜃)) ∧ (𝜒 → 𝜃)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629

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-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)