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Theorem anddi 913
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 911 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑 ∧ (𝜒𝜃)) ∨ (𝜓 ∧ (𝜒𝜃))))
2 andi 910 . . 3 ((𝜑 ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜑𝜃)))
3 andi 910 . . 3 ((𝜓 ∧ (𝜒𝜃)) ↔ ((𝜓𝜒) ∨ (𝜓𝜃)))
42, 3orbi12i 543 . 2 (((𝜑 ∧ (𝜒𝜃)) ∨ (𝜓 ∧ (𝜒𝜃))) ↔ (((𝜑𝜒) ∨ (𝜑𝜃)) ∨ ((𝜓𝜒) ∨ (𝜓𝜃))))
51, 4bitri 264 1 (((𝜑𝜓) ∧ (𝜒𝜃)) ↔ (((𝜑𝜒) ∨ (𝜑𝜃)) ∨ ((𝜓𝜒) ∨ (𝜓𝜃))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wa 384
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 197  df-or 385  df-an 386
This theorem is referenced by:  prnebg  4380  funun  5920  disjxpin  29373  icoreclin  33176  undif3VD  38938
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