Theorem nnexmid 854

Description: Double negation of decidability of a formula. See also comment of nndc 855 to avoid a pitfall that could come from the label "nnexmid". This theorem can also be proved from bj-nnor 16008 as in bj-nndcALT 16032. (Contributed by BJ, 9-Oct-2019.)

Assertion

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

Proof of Theorem nnexmid

Step	Hyp	Ref	Expression
1		pm3.24 697	. 2 ⊢ ¬ (¬ 𝜑 ∧ ¬ ¬ 𝜑)
2		ioran 756	. 2 ⊢ (¬ (𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∧ ¬ ¬ 𝜑))
3	1, 2	mtbir 675	1 ⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nndc  855  dcfromnotnotr  1470  dcfromcon  1471  exmid1stab  4271