Theorem notnotbdc 867

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 624, 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 624	. 2 |- ( ph -> -. -. ph )
2		notnotrdc 838	. 2 |- (DECID ph -> ( -. -. ph -> ph ) )
3	1, 2	impbid2 142	1 |- (DECID ph -> ( ph <-> -. -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  DECID wdc 829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-dc 830

This theorem is referenced by:  con1biidc  872  imordc  892  dfbi3dc  1392  alexdc  1612