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Theorem con1biidc 863
Description: A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
Hypothesis
Ref Expression
con1biidc.1 (DECID 𝜑 → (¬ 𝜑𝜓))
Assertion
Ref Expression
con1biidc (DECID 𝜑 → (¬ 𝜓𝜑))

Proof of Theorem con1biidc
StepHypRef Expression
1 notnotbdc 858 . . 3 (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
2 con1biidc.1 . . . 4 (DECID 𝜑 → (¬ 𝜑𝜓))
32notbid 657 . . 3 (DECID 𝜑 → (¬ ¬ 𝜑 ↔ ¬ 𝜓))
41, 3bitrd 187 . 2 (DECID 𝜑 → (𝜑 ↔ ¬ 𝜓))
54bicomd 140 1 (DECID 𝜑 → (¬ 𝜓𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  DECID wdc 820
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 604  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-dc 821
This theorem is referenced by:  con2biidc  865  necon1abiidc  2387  necon1bbiidc  2388
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