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Theorem anddi 816
Description: Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
anddi (((𝜑𝜓) ∧ (𝜒𝜃)) ↔ (((𝜑𝜒) ∨ (𝜑𝜃)) ∨ ((𝜓𝜒) ∨ (𝜓𝜃))))

Proof of Theorem anddi
StepHypRef Expression
1 andir 814 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑 ∧ (𝜒𝜃)) ∨ (𝜓 ∧ (𝜒𝜃))))
2 andi 813 . . 3 ((𝜑 ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜑𝜃)))
3 andi 813 . . 3 ((𝜓 ∧ (𝜒𝜃)) ↔ ((𝜓𝜒) ∨ (𝜓𝜃)))
42, 3orbi12i 759 . 2 (((𝜑 ∧ (𝜒𝜃)) ∨ (𝜓 ∧ (𝜒𝜃))) ↔ (((𝜑𝜒) ∨ (𝜑𝜃)) ∨ ((𝜓𝜒) ∨ (𝜓𝜃))))
51, 4bitri 183 1 (((𝜑𝜓) ∧ (𝜒𝜃)) ↔ (((𝜑𝜒) ∨ (𝜑𝜃)) ∨ ((𝜓𝜒) ∨ (𝜓𝜃))))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wo 703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  funun  5240  acexmidlemcase  5845  nnm00  6505
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