Theorem xorexmid 1534

Description: Exclusive-or variant of the law of the excluded middle (exmid 900). 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		pm5.19 387	. 2 ⊢ ¬ (𝜑 ↔ ¬ 𝜑)
2		df-xor 1519	. 2 ⊢ ((𝜑 ⊻ ¬ 𝜑) ↔ ¬ (𝜑 ↔ ¬ 𝜑))
3	1, 2	mpbir 232	1 ⊢ (𝜑 ⊻ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 207   ⊻ wxo 1518

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 208  df-xor 1519

This theorem is referenced by: (None)