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Theorem anddi 745
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 743 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑 ∧ (𝜒𝜃)) ∨ (𝜓 ∧ (𝜒𝜃))))
2 andi 742 . . 3 ((𝜑 ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜑𝜃)))
3 andi 742 . . 3 ((𝜓 ∧ (𝜒𝜃)) ↔ ((𝜓𝜒) ∨ (𝜓𝜃)))
42, 3orbi12i 691 . 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:  funun  4972  acexmidlemcase  5535  nnm00  6133
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