Theorem notnotrdc 751

Description: Double negation elimination for a decidable proposition. The converse, notnot 559, 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 743	. . 3 |- (DECID ph <-> ( ph \/ -. ph ) )
2		orcom 647	. . 3 |- ( ( ph \/ -. ph ) <-> ( -. ph \/ ph ) )
3	1, 2	bitri 173	. 2 |- (DECID ph <-> ( -. ph \/ ph ) )
4		pm2.53 641	. 2 |- ( ( -. ph \/ ph ) -> ( -. -. ph -> ph ) )
5	3, 4	sylbi 114	1 |- (DECID ph -> ( -. -. ph -> ph ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  dcimpstab  752  notnotbdc  766  condandc  775  pm2.13dc  779  pm2.54dc  790