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Theorem xordi 1045
 Description: Conjunction distributes over exclusive-or, using ¬ (𝜑 ↔ 𝜓) to express exclusive-or. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. This is not necessarily true in intuitionistic logic, though anxordi 1652 does hold in it. (Contributed by NM, 3-Oct-2008.)
Assertion
Ref Expression
xordi ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))

Proof of Theorem xordi
StepHypRef Expression
1 annim 394 . 2 ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ (𝜑 → (𝜓𝜒)))
2 pm5.32 569 . 2 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
31, 2xchbinx 326 1 ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386 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 199  df-an 387 This theorem is referenced by:  anxordi  1652
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