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Theorem xordi 973
Description: Conjunction distributes over exclusive-or, using ¬ (𝜑𝜓) 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 1592 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 440 . 2 ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ (𝜑 → (𝜓𝜒)))
2 pm5.32 671 . 2 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
31, 2xchbinx 323 1 ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383
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-an 385
This theorem is referenced by:  anxordi  1592
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