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

Theorem notnotrd 135
Description: Deduction associated with notnotr 132 and notnotri 133. Double negation elimination rule. A translation of the natural deduction rule ¬ ¬ C , Γ¬ ¬ 𝜓 ⇒ Γ𝜓; see natded 28340. This is Definition NNC in [Pfenning] p. 17. This rule is valid in classical logic (our logic), but not in intuitionistic logic. (Contributed by DAW, 8-Feb-2017.)
Hypothesis
Ref Expression
notnotrd.1 (𝜑 → ¬ ¬ 𝜓)
Assertion
Ref Expression
notnotrd (𝜑𝜓)

Proof of Theorem notnotrd
StepHypRef Expression
1 notnotrd.1 . 2 (𝜑 → ¬ ¬ 𝜓)
2 notnotr 132 . 2 (¬ ¬ 𝜓𝜓)
31, 2syl 17 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem is referenced by:  condan  818  ecase2d  1029  efald  1563  necon1ai  2961  supgtoreq  9007  konigthlem  10068  indpi  10407  sqrmo  14701  2sqcoprm  26171  axtgupdim2  26417  ncoltgdim2  26511  ex-natded5.13  28352  bnj1204  32563  knoppndvlem10  34339  supxrgere  42410  supxrgelem  42414  supxrge  42415  iccdifprioo  42594  icccncfext  42970  stirlinglem5  43161  sge0repnf  43466  sge0split  43489  nnfoctbdjlem  43535  nabctnabc  43965
  Copyright terms: Public domain W3C validator