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Theorem anxordi 1648
Description: Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 1040 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 1040 . 2 ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
2 df-xor 1634 . . 3 ((𝜓𝜒) ↔ ¬ (𝜓𝜒))
32anbi2i 616 . 2 ((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ¬ (𝜓𝜒)))
4 df-xor 1634 . 2 (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
51, 3, 43bitr4i 294 1 ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384  wxo 1633
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 198  df-an 385  df-xor 1634
This theorem is referenced by: (None)
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