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Theorem orimdi 887
Description: Disjunction distributes over implication. (Contributed by Wolf Lammen, 5-Jan-2013.)
Assertion
Ref Expression
orimdi ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))

Proof of Theorem orimdi
StepHypRef Expression
1 imdi 376 . 2 ((¬ 𝜑 → (𝜓𝜒)) ↔ ((¬ 𝜑𝜓) → (¬ 𝜑𝜒)))
2 df-or 383 . 2 ((𝜑 ∨ (𝜓𝜒)) ↔ (¬ 𝜑 → (𝜓𝜒)))
3 df-or 383 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
4 df-or 383 . . 3 ((𝜑𝜒) ↔ (¬ 𝜑𝜒))
53, 4imbi12i 338 . 2 (((𝜑𝜓) → (𝜑𝜒)) ↔ ((¬ 𝜑𝜓) → (¬ 𝜑𝜒)))
61, 2, 53bitr4i 290 1 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381
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
This theorem is referenced by:  pm2.76  888  pm2.85  893
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