Theorem notnotrd 128

Description: Deduction associated with notnotr 125 and notnotri 126. Double negation elimination rule. A translation of the natural deduction rule ¬ ¬ C , Γ⊢ ¬ ¬ 𝜓 ⇒ Γ⊢ 𝜓; see natded 27260. 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 125	. 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  835  efald  1504  necon1ai  2821  supgtoreq  8376  konigthlem  9390  indpi  9729  sqrmo  13992  axtgupdim2  25370  ncoltgdim2  25460  ex-natded5.13  27272  2sqcoprm  29647  bnj1204  31080  knoppndvlem10  32512  supxrgere  39549  supxrgelem  39553  supxrge  39554  iccdifprioo  39742  icccncfext  40100  stirlinglem5  40295  sge0repnf  40603  sge0split  40626  nnfoctbdjlem  40672  nabctnabc  41098