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Theorem nndc 6264
Description: Double negation of excluded middle. 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 6263 . 2 ¬ ¬ (φ ¬ φ)
2 df-dc 734 . . 3 (DECID φ ↔ (φ ¬ φ))
32notbii 581 . 2 DECID φ ↔ ¬ (φ ¬ φ))
41, 3mtbir 583 1 ¬ ¬ DECID φ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wo 616  DECID wdc 733
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 532  ax-in2 533  ax-io 617
This theorem depends on definitions:  df-bi 110  df-dc 734
This theorem is referenced by:  dcdc  6266
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