Theorem pm4.79 1001

Description: Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.)

Assertion

Ref	Expression
pm4.79	⊢ (((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)) ↔ ((𝜓 ∧ 𝜒) → 𝜑))

Proof of Theorem pm4.79

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ ((𝜓 → 𝜑) → (𝜓 → 𝜑))
2		id 22	. . 3 ⊢ ((𝜒 → 𝜑) → (𝜒 → 𝜑))
3	1, 2	jaoa 953	. 2 ⊢ (((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)) → ((𝜓 ∧ 𝜒) → 𝜑))
4		simplim 167	. . . 4 ⊢ (¬ (𝜓 → 𝜑) → 𝜓)
5		pm3.3 449	. . . 4 ⊢ (((𝜓 ∧ 𝜒) → 𝜑) → (𝜓 → (𝜒 → 𝜑)))
6	4, 5	syl5 34	. . 3 ⊢ (((𝜓 ∧ 𝜒) → 𝜑) → (¬ (𝜓 → 𝜑) → (𝜒 → 𝜑)))
7	6	orrd 860	. 2 ⊢ (((𝜓 ∧ 𝜒) → 𝜑) → ((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)))
8	3, 7	impbii 208	1 ⊢ (((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)) ↔ ((𝜓 ∧ 𝜒) → 𝜑))

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:  reuprg  4639  islinindfis  45790