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	$$⊢\left(\phi ⊻\neg \phi \right)$$

Proof

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