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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 30200. 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  817  ecase2d  1028  efald  1555  necon1ai  2963  supgtoreq  9485  konigthlem  10583  indpi  10922  sqrmo  15222  2sqcoprm  27355  axtgupdim2  28262  ncoltgdim2  28356  ex-natded5.13  30212  bnj1204  34579  knoppndvlem10  35932  hashnexinj  41531  supxrgere  44638  supxrgelem  44642  supxrge  44643  iccdifprioo  44824  icccncfext  45198  stirlinglem5  45389  sge0repnf  45697  sge0split  45720  nnfoctbdjlem  45766  nabctnabc  46236
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