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Theorem pm2.18dc 750
 Description: Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called the Law of Clavius). Intuitionistically it requires a decidability assumption, but compare with pm2.01 546 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)
Assertion
Ref Expression
pm2.18dc (DECID 𝜑 → ((¬ 𝜑𝜑) → 𝜑))

Proof of Theorem pm2.18dc
StepHypRef Expression
1 pm2.21 547 . . . 4 𝜑 → (𝜑 → ¬ (¬ 𝜑𝜑)))
21a2i 11 . . 3 ((¬ 𝜑𝜑) → (¬ 𝜑 → ¬ (¬ 𝜑𝜑)))
3 condc 749 . . 3 (DECID 𝜑 → ((¬ 𝜑 → ¬ (¬ 𝜑𝜑)) → ((¬ 𝜑𝜑) → 𝜑)))
42, 3syl5 28 . 2 (DECID 𝜑 → ((¬ 𝜑𝜑) → ((¬ 𝜑𝜑) → 𝜑)))
54pm2.43d 44 1 (DECID 𝜑 → ((¬ 𝜑𝜑) → 𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  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:  pm4.81dc  814
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