Theorem xor2 1618

Description: Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
xor2	⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))

Proof of Theorem xor2

Step	Hyp	Ref	Expression
1		df-xor 1613	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
2		nbi2 1001	. 2 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
3	1, 2	bitri 264	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 382   ∨ wo 836   ⊻ 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-or 837  df-xor 1613

This theorem is referenced by:  xoror  1619  xornan  1620  cador  1695  saddisjlem  15394  ifpdfxor  38358  dfxor4  38584  nanorxor  39030