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

Theorem anxordi 1512
Description: Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 1012 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 1012 . 2 ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
2 df-xor 1498 . . 3 ((𝜓𝜒) ↔ ¬ (𝜓𝜒))
32anbi2i 622 . 2 ((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ¬ (𝜓𝜒)))
4 df-xor 1498 . 2 (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
51, 3, 43bitr4i 304 1 ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  wxo 1497
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 208  df-an 397  df-xor 1498
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator