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Theorem notnotrdc 751
 Description: Double negation elimination for a decidable proposition. The converse, notnot 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
notnotrdc (DECID 𝜑 → (¬ ¬ 𝜑𝜑))

Proof of Theorem notnotrdc
StepHypRef Expression
1 df-dc 743 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orcom 647 . . 3 ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑𝜑))
31, 2bitri 173 . 2 (DECID 𝜑 ↔ (¬ 𝜑𝜑))
4 pm2.53 641 . 2 ((¬ 𝜑𝜑) → (¬ ¬ 𝜑𝜑))
53, 4sylbi 114 1 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 629  DECID wdc 742 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 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  dcimpstab  752  notnotbdc  766  condandc  775  pm2.13dc  779  pm2.54dc  790
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