Theorem nnexmid 10570

Description: Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but, of course, does not prove excluded middle) for any formula. (Contributed by BJ, 9-Oct-2019.)

Assertion

Ref	Expression
nnexmid	⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑)

Proof of Theorem nnexmid

Step	Hyp	Ref	Expression
1		pm3.24 659	. 2 ⊢ ¬ (¬ 𝜑 ∧ ¬ ¬ 𝜑)
2		ioran 701	. 2 ⊢ (¬ (𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∧ ¬ ¬ 𝜑))
3	1, 2	mtbir 628	1 ⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 102   ∨ wo 661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  nndc  10571