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Theorem con1dc 753
 Description: Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.)
Assertion
Ref Expression
con1dc (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))

Proof of Theorem con1dc
StepHypRef Expression
1 notnot 559 . . 3 (𝜓 → ¬ ¬ 𝜓)
21imim2i 12 . 2 ((¬ 𝜑𝜓) → (¬ 𝜑 → ¬ ¬ 𝜓))
3 condc 749 . 2 (DECID 𝜑 → ((¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓𝜑)))
42, 3syl5 28 1 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
 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-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  impidc  755  simplimdc  757  con1biimdc  767  con1bdc  772  pm3.13dc  866  necon1aidc  2256  necon1bidc  2257  necon1addc  2281  necon1bddc  2282
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