Theorem xor2 1524

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 1519	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
2		nbi2 1023	. 2 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
3	1, 2	bitri 276	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∨ wo 853   ⊻ wxo 1518

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-or 854  df-xor 1519

This theorem is referenced by:  xoror  1525  xornan  1526  cador  1615  saddisjlem  16431  xoromon  35277  wl-df4-3mintru2  37856  ifpdfxor  43938  dfxor4  44217  nanorxor  44756