Theorem notnotrdc 829

Description: Double negation elimination for a decidable proposition. The converse, notnot 619, holds for all propositions, not just decidable ones. This is Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability condition added. (Contributed by Jim Kingdon, 11-Mar-2018.)

Assertion

Ref	Expression
notnotrdc	|- (DECID ph -> ( -. -. ph -> ph ) )

Proof of Theorem notnotrdc

Step	Hyp	Ref	Expression
1		df-dc 821	. . 3 |- (DECID ph <-> ( ph \/ -. ph ) )
2		orcom 718	. . 3 |- ( ( ph \/ -. ph ) <-> ( -. ph \/ ph ) )
3	1, 2	bitri 183	. 2 |- (DECID ph <-> ( -. ph \/ ph ) )
4		pm2.53 712	. 2 |- ( ( -. ph \/ ph ) -> ( -. -. ph -> ph ) )
5	3, 4	sylbi 120	1 |- (DECID ph -> ( -. -. ph -> ph ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 698  DECID wdc 820

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-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 821

This theorem is referenced by:  dcstab  830  notnotbdc  858  condandc  867  pm2.13dc  871  pm2.54dc  877  mkvprop  7040