Theorem bj-dcdc 13640

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
bj-dcdc	⊢ (DECID DECID 𝜑 ↔ DECID 𝜑)

Proof of Theorem bj-dcdc

Step	Hyp	Ref	Expression
1		nndc 841	. 2 ⊢ ¬ ¬ DECID 𝜑
2		bj-nnbidc 13638	. 2 ⊢ (¬ ¬ DECID 𝜑 → (DECID DECID 𝜑 ↔ DECID 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (DECID DECID 𝜑 ↔ DECID 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 104  DECID wdc 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by: (None)