Theorem nndc 9900

Description: Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but, of course, does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.)

Assertion

Ref	Expression
nndc	⊢ ¬ ¬ DECID 𝜑

Proof of Theorem nndc

Step	Hyp	Ref	Expression
1		nnexmid 9899	. 2 ⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑)
2		df-dc 743	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
3	2	notbii 594	. 2 ⊢ (¬ DECID 𝜑 ↔ ¬ (𝜑 ∨ ¬ 𝜑))
4	1, 3	mtbir 596	1 ⊢ ¬ ¬ DECID 𝜑

Syntax hints:  ¬ wn 3   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  dcdc  9901