Theorem notnotrd 131

Description: Deduction associated with notnotr 128 and notnotri 129. Double negation elimination rule. A translation of the natural deduction rule ¬ ¬ C , Γ⊢ ¬ ¬ 𝜓 ⇒ Γ⊢ 𝜓; see natded 27814. 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 128	. 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  852  efald  1678  necon1ai  3026  supgtoreq  8651  konigthlem  9712  indpi  10051  sqrmo  14376  axtgupdim2  25790  ncoltgdim2  25884  ex-natded5.13  27826  2sqcoprm  30188  bnj1204  31622  knoppndvlem10  33039  supxrgere  40344  supxrgelem  40348  supxrge  40349  iccdifprioo  40536  icccncfext  40893  stirlinglem5  41087  sge0repnf  41392  sge0split  41415  nnfoctbdjlem  41461  nabctnabc  41890