Theorem xorexmid

Description: Exclusive-or variant of the law of the excluded middle ( exmid ). This statement is ancient, going back to at least Stoic logic. This statement does not necessarily hold in intuitionistic logic. (Contributed by David A. Wheeler, 23-Feb-2019)

	Ref	Expression
Assertion	xorexmid	⊢ ( 𝜑 ⊻ ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		pm5.19	⊢ ¬ ( 𝜑 ↔ ¬ 𝜑 )
2		df-xor	⊢ ( ( 𝜑 ⊻ ¬ 𝜑 ) ↔ ¬ ( 𝜑 ↔ ¬ 𝜑 ) )
3	1 2	mpbir	⊢ ( 𝜑 ⊻ ¬ 𝜑 )