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Theorem con1dc 841
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 618 . . 3 (𝜓 → ¬ ¬ 𝜓)
21imim2i 12 . 2 ((¬ 𝜑𝜓) → (¬ 𝜑 → ¬ ¬ 𝜓))
3 condc 838 . 2 (DECID 𝜑 → ((¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓𝜑)))
42, 3syl5 32 1 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
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:  impidc  843  simplimdc  845  con1biimdc  858  con1bdc  863  pm3.13dc  943  necon1aidc  2359  necon1bidc  2360  necon1addc  2384  necon1bddc  2385  phiprmpw  11904
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