 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  anxordi Structured version   Visualization version   GIF version

Theorem anxordi 1627
 Description: Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 1002 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 1002 . 2 ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
2 df-xor 1613 . . 3 ((𝜓𝜒) ↔ ¬ (𝜓𝜒))
32anbi2i 609 . 2 ((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ¬ (𝜓𝜒)))
4 df-xor 1613 . 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 382   ⊻ wxo 1612 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 383  df-xor 1613 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator