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Theorem notnotrdc 762
Description: Double negation elimination for a decidable proposition. The converse, notnot 569, 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 754 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orcom 657 . . 3 ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑𝜑))
31, 2bitri 177 . 2 (DECID 𝜑 ↔ (¬ 𝜑𝜑))
4 pm2.53 651 . 2 ((¬ 𝜑𝜑) → (¬ ¬ 𝜑𝜑))
53, 4sylbi 118 1 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  dcimpstab  763  notnotbdc  777  condandc  786  pm2.13dc  790  pm2.54dc  801
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