Theorem notnotrd 134

Description: Deduction associated with notnotr 131 and notnotri 132. Double negation elimination rule. A translation of the natural deduction rule ¬ ¬ C , Γ⊢ ¬ ¬ 𝜓 ⇒ Γ⊢ 𝜓; see natded 30691. 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 131	. 2 ⊢ (¬ ¬ 𝜓 → 𝜓)
3	1, 2	syl 18	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  829  ecase2d  1045  efald  1588  necon1ai  2991  supgtoreq  9427  konigthlem  10549  indpi  10888  sqrmo  15298  2sqcoprm  27561  axtgupdim2  28702  ncoltgdim2  28796  ex-natded5.13  30703  bnj1204  35341  knoppndvlem10  36995  hashnexinj  42780  supxrgere  45936  supxrgelem  45940  supxrge  45941  iccdifprioo  46119  icccncfext  46488  stirlinglem5  46679  sge0repnf  46987  sge0split  47010  nnfoctbdjlem  47056  nabctnabc  47552