Theorem nnexmid 835

Description: Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but does not prove excluded middle) for any formula. Can also be proved quickly from bj-nnor 12935 as in bj-nndcALT 12952. (Contributed by BJ, 9-Oct-2019.)

Assertion

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

Proof of Theorem nnexmid

Step	Hyp	Ref	Expression
1		pm3.24 682	. 2 ⊢ ¬ (¬ 𝜑 ∧ ¬ ¬ 𝜑)
2		ioran 741	. 2 ⊢ (¬ (𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∧ ¬ ¬ 𝜑))
3	1, 2	mtbir 660	1 ⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  nndc  836  exmid1stab  13184