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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 30432. 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  1031  efald  1558  necon1ai  2966  supgtoreq  9508  konigthlem  10606  indpi  10945  sqrmo  15287  2sqcoprm  27494  axtgupdim2  28494  ncoltgdim2  28588  ex-natded5.13  30444  bnj1204  35005  knoppndvlem10  36504  hashnexinj  42110  supxrgere  45283  supxrgelem  45287  supxrge  45288  iccdifprioo  45469  icccncfext  45843  stirlinglem5  46034  sge0repnf  46342  sge0split  46365  nnfoctbdjlem  46411  nabctnabc  46881
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