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Theorem orddi 733
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 730 . 2 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑 ∨ (𝜒𝜃)) ∧ (𝜓 ∨ (𝜒𝜃))))
2 ordi 729 . . 3 ((𝜑 ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜑𝜃)))
3 ordi 729 . . 3 ((𝜓 ∨ (𝜒𝜃)) ↔ ((𝜓𝜒) ∧ (𝜓𝜃)))
42, 3anbi12i 433 . 2 (((𝜑 ∨ (𝜒𝜃)) ∧ (𝜓 ∨ (𝜒𝜃))) ↔ (((𝜑𝜒) ∧ (𝜑𝜃)) ∧ ((𝜓𝜒) ∧ (𝜓𝜃))))
51, 4bitri 173 1 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ (((𝜑𝜒) ∧ (𝜑𝜃)) ∧ ((𝜓𝜒) ∧ (𝜓𝜃))))
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  wo 629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  prneimg  3545
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