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Theorem orddi 909
 Description: Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
orddi (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ (((𝜑𝜒) ∧ (𝜑𝜃)) ∧ ((𝜓𝜒) ∧ (𝜓𝜃))))

Proof of Theorem orddi
StepHypRef Expression
1 ordir 905 . 2 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑 ∨ (𝜒𝜃)) ∧ (𝜓 ∨ (𝜒𝜃))))
2 ordi 904 . . 3 ((𝜑 ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜑𝜃)))
3 ordi 904 . . 3 ((𝜓 ∨ (𝜒𝜃)) ↔ ((𝜓𝜒) ∧ (𝜓𝜃)))
42, 3anbi12i 729 . 2 (((𝜑 ∨ (𝜒𝜃)) ∧ (𝜓 ∨ (𝜒𝜃))) ↔ (((𝜑𝜒) ∧ (𝜑𝜃)) ∧ ((𝜓𝜒) ∧ (𝜓𝜃))))
51, 4bitri 263 1 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ (((𝜑𝜒) ∧ (𝜑𝜃)) ∧ ((𝜓𝜒) ∧ (𝜓𝜃))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∨ wo 382   ∧ wa 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 196  df-or 384  df-an 385 This theorem is referenced by:  prneimg  4328  wl-cases2-dnf  32474
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