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Theorem dcdc 9216
Description: Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)
Assertion
Ref Expression
dcdc (DECID DECID φDECID φ)

Proof of Theorem dcdc
StepHypRef Expression
1 df-dc 742 . 2 (DECID DECID φ ↔ (DECID φ ¬ DECID φ))
2 nndc 9215 . . 3 ¬ ¬ DECID φ
32biorfi 664 . 2 (DECID φ ↔ (DECID φ ¬ DECID φ))
41, 3bitr4i 176 1 (DECID DECID φDECID φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98   wo 628  DECID wdc 741
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 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by: (None)
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