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Theorem orddi 744
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 741 . 2 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑 ∨ (𝜒𝜃)) ∧ (𝜓 ∨ (𝜒𝜃))))
2 ordi 740 . . 3 ((𝜑 ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜑𝜃)))
3 ordi 740 . . 3 ((𝜓 ∨ (𝜒𝜃)) ↔ ((𝜓𝜒) ∧ (𝜓𝜃)))
42, 3anbi12i 441 . 2 (((𝜑 ∨ (𝜒𝜃)) ∧ (𝜓 ∨ (𝜒𝜃))) ↔ (((𝜑𝜒) ∧ (𝜑𝜃)) ∧ ((𝜓𝜒) ∧ (𝜓𝜃))))
51, 4bitri 177 1 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ (((𝜑𝜒) ∧ (𝜑𝜃)) ∧ ((𝜓𝜒) ∧ (𝜓𝜃))))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wo 639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  prneimg  3572
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