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Theorem dcbii 785
Description: The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
Hypothesis
Ref Expression
dcbii.1 (𝜑𝜓)
Assertion
Ref Expression
dcbii (DECID 𝜑DECID 𝜓)

Proof of Theorem dcbii
StepHypRef Expression
1 dcbii.1 . . 3 (𝜑𝜓)
21notbii 629 . . 3 𝜑 ↔ ¬ 𝜓)
31, 2orbi12i 716 . 2 ((𝜑 ∨ ¬ 𝜑) ↔ (𝜓 ∨ ¬ 𝜓))
4 df-dc 781 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5 df-dc 781 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
63, 4, 53bitr4i 210 1 (DECID 𝜑DECID 𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103  wo 664  DECID wdc 780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by:  dcbi  882  dcned  2261  dfrex2dc  2371  euxfr2dc  2800  exmidexmid  4031  exfzdc  9647  nninfsellemdc  11857
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