Theorem xor2

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

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

Proof

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