Theorem xorexmid 1472

Description: Exclusive-or variant of the law of the excluded middle (exmid 430). 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.)

Assertion

Ref	Expression
xorexmid	⊢ (𝜑 ⊻ ¬ 𝜑)

Proof of Theorem xorexmid

Step	Hyp	Ref	Expression
1		pm5.19 374	. 2 ⊢ ¬ (𝜑 ↔ ¬ 𝜑)
2		df-xor 1457	. 2 ⊢ ((𝜑 ⊻ ¬ 𝜑) ↔ ¬ (𝜑 ↔ ¬ 𝜑))
3	1, 2	mpbir 220	1 ⊢ (𝜑 ⊻ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 195   ⊻ wxo 1456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-xor 1457

This theorem is referenced by: (None)