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Theorem pm2.18dc 840
 Description: Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called Clavius law). Intuitionistically it requires a decidability assumption, but compare with pm2.01 605 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 pm2.21 606 . . . 4 𝜑 → (𝜑 → ¬ (¬ 𝜑𝜑)))
21a2i 11 . . 3 ((¬ 𝜑𝜑) → (¬ 𝜑 → ¬ (¬ 𝜑𝜑)))
3 condc 838 . . 3 (DECID 𝜑 → ((¬ 𝜑 → ¬ (¬ 𝜑𝜑)) → ((¬ 𝜑𝜑) → 𝜑)))
42, 3syl5 32 . 2 (DECID 𝜑 → ((¬ 𝜑𝜑) → ((¬ 𝜑𝜑) → 𝜑)))
54pm2.43d 50 1 (DECID 𝜑 → ((¬ 𝜑𝜑) → 𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 819 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-in1 603  ax-in2 604  ax-io 698 This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820 This theorem is referenced by:  pm4.81dc  893
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