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Theorem orbidi 972
Description: Disjunction distributes over the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384. (Contributed by NM, 8-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Feb-2013.)
Assertion
Ref Expression
orbidi ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))

Proof of Theorem orbidi
StepHypRef Expression
1 pm5.74 259 . 2 ((¬ 𝜑 → (𝜓𝜒)) ↔ ((¬ 𝜑𝜓) ↔ (¬ 𝜑𝜒)))
2 df-or 385 . 2 ((𝜑 ∨ (𝜓𝜒)) ↔ (¬ 𝜑 → (𝜓𝜒)))
3 df-or 385 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
4 df-or 385 . . 3 ((𝜑𝜒) ↔ (¬ 𝜑𝜒))
53, 4bibi12i 329 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ↔ (¬ 𝜑𝜒)))
61, 2, 53bitr4i 292 1 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383
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 385
This theorem is referenced by:  pm5.7  974
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