Theorem dcdc 10572

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 776	. 2 |- (DECID DECID ph <-> (DECID ph \/ -. DECID ph ) )
2		nndc 10571	. . 3 |- -. -. DECID ph
3	2	biorfi 697	. 2 |- (DECID ph <-> (DECID ph \/ -. DECID ph ) )
4	1, 3	bitr4i 185	1 |- (DECID DECID ph <-> DECID ph )

Syntax hints:   -. wn 3   <-> wb 103   \/ wo 661  DECID wdc 775

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 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by: (None)