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 𝜑 ↔ DECID 𝜑)

Proof of Theorem dcdc

Step	Hyp	Ref	Expression
1		df-dc 777	. 2 ⊢ (DECID DECID 𝜑 ↔ (DECID 𝜑 ∨ ¬ DECID 𝜑))
2		nndc 10722	. . 3 ⊢ ¬ ¬ DECID 𝜑
3	2	biorfi 698	. 2 ⊢ (DECID 𝜑 ↔ (DECID 𝜑 ∨ ¬ DECID 𝜑))
4	1, 3	bitr4i 185	1 ⊢ (DECID DECID 𝜑 ↔ DECID 𝜑)

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)