Theorem nbi2

Description: Two ways to express "exclusive or". (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 24-Jan-2013)

	Ref	Expression
Assertion	nbi2	⊢ ( ¬ ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xor3	⊢ ( ¬ ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜑 ↔ ¬ 𝜓 ) )
2		pm5.17	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ↔ ¬ 𝜓 ) )
3	1 2	bitr4i	⊢ ( ¬ ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) )