Theorem pm4.78 931

Description: Implication distributes over disjunction. Theorem *4.78 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)

Assertion

Ref	Expression
pm4.78	⊢ (((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒)))

Proof of Theorem pm4.78

Step	Hyp	Ref	Expression
1		orordi 925	. 2 ⊢ ((¬ 𝜑 ∨ (𝜓 ∨ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ (¬ 𝜑 ∨ 𝜒)))
2		imor 849	. 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (¬ 𝜑 ∨ (𝜓 ∨ 𝜒)))
3		imor 849	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
4		imor 849	. . 3 ⊢ ((𝜑 → 𝜒) ↔ (¬ 𝜑 ∨ 𝜒))
5	3, 4	orbi12i 911	. 2 ⊢ (((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ (¬ 𝜑 ∨ 𝜒)))
6	1, 2, 5	3bitr4ri 303	1 ⊢ (((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 843

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-or 844

This theorem is referenced by: (None)