Theorem notnotbdc 799

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 591, 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 591	. 2 |- ( ph -> -. -. ph )
2		notnotrdc 784	. 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 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:  con1biidc  804  imandc  819  imordc  829  dfbi3dc  1328  alexdc  1550