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

Theorem notnotrd 133
Description: Deduction associated with notnotr 130 and notnotri 131. Double negation elimination rule. A translation of the natural deduction rule ¬ ¬ C , Γ¬ ¬ 𝜓 ⇒ Γ𝜓; see natded 29920. 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 130 . 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  815  ecase2d  1027  efald  1561  necon1ai  2967  supgtoreq  9468  konigthlem  10566  indpi  10905  sqrmo  15203  2sqcoprm  27171  axtgupdim2  27986  ncoltgdim2  28080  ex-natded5.13  29932  bnj1204  34318  knoppndvlem10  35701  supxrgere  44343  supxrgelem  44347  supxrge  44348  iccdifprioo  44529  icccncfext  44903  stirlinglem5  45094  sge0repnf  45402  sge0split  45425  nnfoctbdjlem  45471  nabctnabc  45941
  Copyright terms: Public domain W3C validator