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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 30422. 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  818  ecase2d  1032  efald  1561  necon1ai  2968  supgtoreq  9510  konigthlem  10608  indpi  10947  sqrmo  15290  2sqcoprm  27479  axtgupdim2  28479  ncoltgdim2  28573  ex-natded5.13  30434  bnj1204  35026  knoppndvlem10  36522  hashnexinj  42129  supxrgere  45344  supxrgelem  45348  supxrge  45349  iccdifprioo  45529  icccncfext  45902  stirlinglem5  46093  sge0repnf  46401  sge0split  46424  nnfoctbdjlem  46470  nabctnabc  46943
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