Theorem notnotrdc 855

Description: Double negation elimination for a decidable proposition. The converse, notnot 638, 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 847	. . 3 |- (DECID ph <-> ( ph \/ -. ph ) )
2		orcom 740	. . 3 |- ( ( ph \/ -. ph ) <-> ( -. ph \/ ph ) )
3	1, 2	bitri 184	. 2 |- (DECID ph <-> ( -. ph \/ ph ) )
4		pm2.53 734	. 2 |- ( ( -. ph \/ ph ) -> ( -. -. ph -> ph ) )
5	3, 4	sylbi 121	1 |- (DECID ph -> ( -. -. ph -> ph ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 720  DECID wdc 846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721

This theorem depends on definitions:  df-bi 117  df-dc 847

This theorem is referenced by:  dcstab  856  notnotbdc  884  condandc  893  pm2.13dc  897  pm2.54dc  903  mkvprop  7488  netap  7610  bitsfzo  12700  ballotfilemic  13228  exmidnotnotr  16949