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Theorem con1dc 764
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 569 . . 3 (𝜓 → ¬ ¬ 𝜓)
21imim2i 12 . 2 ((¬ 𝜑𝜓) → (¬ 𝜑 → ¬ ¬ 𝜓))
3 condc 760 . 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 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  impidc  766  simplimdc  768  con1biimdc  778  con1bdc  783  pm3.13dc  877  necon1aidc  2271  necon1bidc  2272  necon1addc  2296  necon1bddc  2297
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