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Theorem pm2.1dc 823
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 821 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orcom 718 . . 3 ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑𝜑))
31, 2bitri 183 . 2 (DECID 𝜑 ↔ (¬ 𝜑𝜑))
43biimpi 119 1 (DECID 𝜑 → (¬ 𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 698  DECID wdc 820
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-io 699
This theorem depends on definitions:  df-bi 116  df-dc 821
This theorem is referenced by:  pm2.6dc  848  rabrsndc  3597
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