Theorem xorexmid 1524

Description: Exclusive-or variant of the law of the excluded middle (exmid 892). 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		pm5.19 388	. 2 ⊢ ¬ (𝜑 ↔ ¬ 𝜑)
2		df-xor 1507	. 2 ⊢ ((𝜑 ⊻ ¬ 𝜑) ↔ ¬ (𝜑 ↔ ¬ 𝜑))
3	1, 2	mpbir 230	1 ⊢ (𝜑 ⊻ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 205   ⊻ wxo 1506

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 206  df-xor 1507

This theorem is referenced by: (None)