Theorem dcdc 9236

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 ph <-> DECID ph)

Proof of Theorem dcdc

Step	Hyp	Ref	Expression
1		df-dc 742	. 2 |- (DECID DECID ph <-> (DECID ph \/ -. DECID ph))
2		nndc 9235	. . 3 |- -. -. DECID ph
3	2	biorfi 664	. 2 |- (DECID ph <-> (DECID ph \/ -. DECID ph))
4	1, 3	bitr4i 176	1 |- (DECID DECID ph <-> DECID ph)

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)