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 ⇒ Γ⊢ 𝜓; see natded 29694. 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	⊢ (𝜑 → 𝜓)

Proof of Theorem notnotrd

Step	Hyp	Ref	Expression
1		notnotrd.1	. 2 ⊢ (𝜑 → ¬ ¬ 𝜓)
2		notnotr 130	. 2 ⊢ (¬ ¬ 𝜓 → 𝜓)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝜓)

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  816  ecase2d  1028  efald  1562  necon1ai  2968  supgtoreq  9467  konigthlem  10565  indpi  10904  sqrmo  15200  2sqcoprm  26945  axtgupdim2  27760  ncoltgdim2  27854  ex-natded5.13  29706  bnj1204  34092  knoppndvlem10  35483  supxrgere  44122  supxrgelem  44126  supxrge  44127  iccdifprioo  44308  icccncfext  44682  stirlinglem5  44873  sge0repnf  45181  sge0split  45204  nnfoctbdjlem  45250  nabctnabc  45720