Theorem xor2 1494

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 1489	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
2		nbi2 998	. 2 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
3	1, 2	bitri 267	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 198   ∧ wa 387   ∨ wo 833   ⊻ wxo 1488

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 199  df-an 388  df-or 834  df-xor 1489

This theorem is referenced by:  xoror  1495  xornan  1496  cador  1571  saddisjlem  15673  ifpdfxor  39255  dfxor4  39480  nanorxor  40059