Theorem notnotbdc 804

Description: Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 594, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.)

Assertion

Ref	Expression
notnotbdc	|- (DECID ph -> ( ph <-> -. -. ph ) )

Proof of Theorem notnotbdc

Step	Hyp	Ref	Expression
1		notnot 594	. 2 |- ( ph -> -. -. ph )
2		notnotrdc 789	. 2 |- (DECID ph -> ( -. -. ph -> ph ) )
3	1, 2	impbid2 141	1 |- (DECID ph -> ( ph <-> -. -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103  DECID wdc 780

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 579  ax-in2 580  ax-io 665

This theorem depends on definitions:  df-bi 115  df-dc 781

This theorem is referenced by:  con1biidc  809  imandc  824  imordc  834  dfbi3dc  1333  alexdc  1555