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Theorem notnotrdc 811
Description: Double negation elimination for a decidable proposition. The converse, notnot 601, 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 803 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orcom 700 . . 3 ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑𝜑))
31, 2bitri 183 . 2 (DECID 𝜑 ↔ (¬ 𝜑𝜑))
4 pm2.53 694 . 2 ((¬ 𝜑𝜑) → (¬ ¬ 𝜑𝜑))
53, 4sylbi 120 1 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 680  DECID wdc 802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-dc 803
This theorem is referenced by:  dcstab  812  notnotbdc  840  condandc  849  pm2.13dc  853  pm2.54dc  859  mkvprop  6998
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