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Theorem pm2.1dc 779
Description: Commuted law of the excluded middle for a decidable proposition. Based on theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by Jim Kingdon, 25-Mar-2018.)
Assertion
Ref Expression
pm2.1dc (DECID 𝜑 → (¬ 𝜑𝜑))

Proof of Theorem pm2.1dc
StepHypRef Expression
1 df-dc 777 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orcom 680 . . 3 ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑𝜑))
31, 2bitri 182 . 2 (DECID 𝜑 ↔ (¬ 𝜑𝜑))
43biimpi 118 1 (DECID 𝜑 → (¬ 𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 662  DECID wdc 776
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-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  pm2.6dc  793  rabrsndc  3484
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