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Theorem con1biidc 782
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 777 . . 3 (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
2 con1biidc.1 . . . 4 (DECID 𝜑 → (¬ 𝜑𝜓))
32notbid 602 . . 3 (DECID 𝜑 → (¬ ¬ 𝜑 ↔ ¬ 𝜓))
41, 3bitrd 181 . 2 (DECID 𝜑 → (𝜑 ↔ ¬ 𝜓))
54bicomd 133 1 (DECID 𝜑 → (¬ 𝜓𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  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:  con2biidc  784  necon1abiidc  2280  necon1bbiidc  2281
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