Theorem nnexmid 851

Description: Double negation of decidability of a formula. See also comment of nndc 852 to avoid a pitfall that could come from the label "nnexmid". This theorem can also be proved from bj-nnor 15380 as in bj-nndcALT 15404. (Contributed by BJ, 9-Oct-2019.)

Assertion

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

Proof of Theorem nnexmid

Step	Hyp	Ref	Expression
1		pm3.24 694	. 2 ⊢ ¬ (¬ 𝜑 ∧ ¬ ¬ 𝜑)
2		ioran 753	. 2 ⊢ (¬ (𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∧ ¬ ¬ 𝜑))
3	1, 2	mtbir 672	1 ⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 709

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nndc  852  dcfromnotnotr  1458  dcfromcon  1459  exmid1stab  4241