Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  nndc GIF version

Theorem nndc 10287
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
StepHypRef Expression
1 nnexmid 10286 . 2 ¬ ¬ (𝜑 ∨ ¬ 𝜑)
2 df-dc 754 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
32notbii 604 . 2 DECID 𝜑 ↔ ¬ (𝜑 ∨ ¬ 𝜑))
41, 3mtbir 606 1 ¬ ¬ DECID 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  dcdc  10288
  Copyright terms: Public domain W3C validator