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Theorem pm4.79 604
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 (((𝜓𝜑) ∨ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))

Proof of Theorem pm4.79
StepHypRef Expression
1 id 22 . . 3 ((𝜓𝜑) → (𝜓𝜑))
2 id 22 . . 3 ((𝜒𝜑) → (𝜒𝜑))
31, 2jaoa 530 . 2 (((𝜓𝜑) ∨ (𝜒𝜑)) → ((𝜓𝜒) → 𝜑))
4 simplim 161 . . . 4 (¬ (𝜓𝜑) → 𝜓)
5 pm3.3 458 . . . 4 (((𝜓𝜒) → 𝜑) → (𝜓 → (𝜒𝜑)))
64, 5syl5 33 . . 3 (((𝜓𝜒) → 𝜑) → (¬ (𝜓𝜑) → (𝜒𝜑)))
76orrd 391 . 2 (((𝜓𝜒) → 𝜑) → ((𝜓𝜑) ∨ (𝜒𝜑)))
83, 7impbii 197 1 (((𝜓𝜑) ∨ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 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 195  df-or 383  df-an 384
This theorem is referenced by:  islinindfis  42030
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