MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  notnotrd Unicode version

Theorem notnotrd 107
Description: Deduction converting double-negation into the original wff, aka the double negation rule. A translation of natural deduction rule  -.  -. -C,  _G |-  -.  -.  ps =>  _G |-  ps; see natded 4. This is definition NNC in [Pfenning] p. 17. This rule is valid in classical logic (which MPE uses), but not intuitionistic logic. (Contributed by DAW, 8-Feb-2017.)
Hypothesis
Ref Expression
notnotrd.1  |-  ( ph  ->  -.  -.  ps )
Assertion
Ref Expression
notnotrd  |-  ( ph  ->  ps )

Proof of Theorem notnotrd
StepHypRef Expression
1 notnotrd.1 . 2  |-  ( ph  ->  -.  -.  ps )
2 notnot2 106 . 2  |-  ( -. 
-.  ps  ->  ps )
31, 2syl 17 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6
This theorem is referenced by:  efald  1330  ex-natded5.13  20755
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
  Copyright terms: Public domain W3C validator