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Theorem notnot2dc 750
Description: Double negation elimination for a decidable proposition. The converse, notnot1 559, holds for all propositions, not just decidable ones. This is Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability condition added. (Contributed by Jim Kingdon, 11-Mar-2018.)
Assertion
Ref Expression
notnot2dc (DECID φ → (¬ ¬ φφ))

Proof of Theorem notnot2dc
StepHypRef Expression
1 df-dc 742 . . 3 (DECID φ ↔ (φ ¬ φ))
2 orcom 646 . . 3 ((φ ¬ φ) ↔ (¬ φ φ))
31, 2bitri 173 . 2 (DECID φ ↔ (¬ φ φ))
4 pm2.53 640 . 2 ((¬ φ φ) → (¬ ¬ φφ))
53, 4sylbi 114 1 (DECID φ → (¬ ¬ φφ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wo 628  DECID wdc 741
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-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  dcimpstab  751  notnotdc  765  condandc  774  pm2.13dc  778  pm2.54dc  789
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