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Theorem dcnbidcnn 7006
Description: The decidability of ¬ φ is equivalent to that of ¬ ¬ φ. (Contributed by BJ, 9-Oct-2019.)
Assertion
Ref Expression
dcnbidcnn (DECID ¬ φDECID ¬ ¬ φ)

Proof of Theorem dcnbidcnn
StepHypRef Expression
1 orcom 634 . . 3 ((¬ ¬ φ ¬ ¬ ¬ φ) ↔ (¬ ¬ ¬ φ ¬ ¬ φ))
2 notnotnot 615 . . . 4 (¬ ¬ ¬ φ ↔ ¬ φ)
32orbi1i 667 . . 3 ((¬ ¬ ¬ φ ¬ ¬ φ) ↔ (¬ φ ¬ ¬ φ))
41, 3bitri 173 . 2 ((¬ ¬ φ ¬ ¬ ¬ φ) ↔ (¬ φ ¬ ¬ φ))
5 df-dc 734 . 2 (DECID ¬ ¬ φ ↔ (¬ ¬ φ ¬ ¬ ¬ φ))
6 df-dc 734 . 2 (DECID ¬ φ ↔ (¬ φ ¬ ¬ φ))
74, 5, 63bitr4ri 202 1 (DECID ¬ φDECID ¬ ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98   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: (None)
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