Theorem pm4.78i 771

Description: Implication distributes over disjunction. One direction of Theorem *4.78 of [WhiteheadRussell] p. 121. The converse holds in classical logic. (Contributed by Jim Kingdon, 15-Jan-2018.)

Assertion

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

Proof of Theorem pm4.78i

Step	Hyp	Ref	Expression
1		orc 701	. . 3 ⊢ (𝜓 → (𝜓 ∨ 𝜒))
2	1	imim2i 12	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜓 ∨ 𝜒)))
3		olc 700	. . 3 ⊢ (𝜒 → (𝜓 ∨ 𝜒))
4	3	imim2i 12	. 2 ⊢ ((𝜑 → 𝜒) → (𝜑 → (𝜓 ∨ 𝜒)))
5	2, 4	jaoi 705	1 ⊢ (((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∨ 𝜒)))

Syntax hints:   → wi 4   ∨ wo 697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)