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Theorem pm4.78 607
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 (((𝜑𝜓) ∨ (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))

Proof of Theorem pm4.78
StepHypRef Expression
1 orordi 553 . 2 ((¬ 𝜑 ∨ (𝜓𝜒)) ↔ ((¬ 𝜑𝜓) ∨ (¬ 𝜑𝜒)))
2 imor 427 . 2 ((𝜑 → (𝜓𝜒)) ↔ (¬ 𝜑 ∨ (𝜓𝜒)))
3 imor 427 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
4 imor 427 . . 3 ((𝜑𝜒) ↔ (¬ 𝜑𝜒))
53, 4orbi12i 544 . 2 (((𝜑𝜓) ∨ (𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∨ (¬ 𝜑𝜒)))
61, 2, 53bitr4ri 293 1 (((𝜑𝜓) ∨ (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382
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 197  df-or 384
This theorem is referenced by: (None)
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