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Theorem dcdc 10723
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 777 . 2 (DECID DECID 𝜑 ↔ (DECID 𝜑 ∨ ¬ DECID 𝜑))
2 nndc 10722 . . 3 ¬ ¬ DECID 𝜑
32biorfi 698 . 2 (DECID 𝜑 ↔ (DECID 𝜑 ∨ ¬ DECID 𝜑))
41, 3bitr4i 185 1 (DECID DECID 𝜑DECID 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103  wo 662  DECID wdc 776
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 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
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