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Theorem anxordi 1477
Description: Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 936 does not necessarily hold, in part because the usual definition of xor is subtly different in intuitionistic logic. (Contributed by David A. Wheeler, 7-Oct-2018.)
Assertion
Ref Expression
anxordi ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒)))

Proof of Theorem anxordi
StepHypRef Expression
1 xordi 936 . 2 ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
2 df-xor 1463 . . 3 ((𝜓𝜒) ↔ ¬ (𝜓𝜒))
32anbi2i 729 . 2 ((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ¬ (𝜓𝜒)))
4 df-xor 1463 . 2 (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
51, 3, 43bitr4i 292 1 ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384  wxo 1462
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 386  df-xor 1463
This theorem is referenced by: (None)
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