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Theorem pm2.18dc 749
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 748 . . 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 741
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 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  pm4.81dc  813
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