Theorem notnotrd 135

Description: Deduction associated with notnotr 132 and notnotri 133. Double negation elimination rule. A translation of the natural deduction rule ¬ ¬ C , Γ⊢ ¬ ¬ 𝜓 ⇒ Γ⊢ 𝜓; see natded 28174. 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 132	. 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  efald  1552  necon1ai  3041  supgtoreq  8926  konigthlem  9982  indpi  10321  sqrmo  14603  2sqcoprm  26003  axtgupdim2  26249  ncoltgdim2  26343  ex-natded5.13  28186  bnj1204  32277  knoppndvlem10  33853  supxrgere  41591  supxrgelem  41595  supxrge  41596  iccdifprioo  41782  icccncfext  42160  stirlinglem5  42354  sge0repnf  42659  sge0split  42682  nnfoctbdjlem  42728  nabctnabc  43158