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Metamath Proof Explorer

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	$⊢ (φ ⊻ \neg φ)$

Proof

Step	Hyp	Ref	Expression
1		pm5.19	$⊢ \neg (φ \leftrightarrow \neg φ)$
2		df-xor	$⊢ (φ ⊻ \neg φ) \leftrightarrow \neg (φ \leftrightarrow \neg φ)$
3	1 2	mpbir	$⊢ (φ ⊻ \neg φ)$