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Theorem con1biidc 771
 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 766 . . 3 (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
2 con1biidc.1 . . . 4 (DECID 𝜑 → (¬ 𝜑𝜓))
32notbid 592 . . 3 (DECID 𝜑 → (¬ ¬ 𝜑 ↔ ¬ 𝜓))
41, 3bitrd 177 . 2 (DECID 𝜑 → (𝜑 ↔ ¬ 𝜓))
54bicomd 129 1 (DECID 𝜑 → (¬ 𝜓𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  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:  con2biidc  773  necon1abiidc  2265  necon1bbiidc  2266
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