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Theorem notnotrdc 789
Description: Double negation elimination for a decidable proposition. The converse, notnot 594, 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
notnotrdc (DECID 𝜑 → (¬ ¬ 𝜑𝜑))

Proof of Theorem notnotrdc
StepHypRef Expression
1 df-dc 781 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orcom 682 . . 3 ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑𝜑))
31, 2bitri 182 . 2 (DECID 𝜑 ↔ (¬ 𝜑𝜑))
4 pm2.53 676 . 2 ((¬ 𝜑𝜑) → (¬ ¬ 𝜑𝜑))
53, 4sylbi 119 1 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 664  DECID wdc 780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by:  dcimpstab  790  notnotbdc  804  condandc  813  pm2.13dc  817  pm2.54dc  828
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