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 30383. 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  817  ecase2d  1031  efald  1562  necon1ai  2955  supgtoreq  9355  konigthlem  10459  indpi  10798  sqrmo  15158  2sqcoprm  27373  axtgupdim2  28449  ncoltgdim2  28543  ex-natded5.13  30395  bnj1204  35024  knoppndvlem10  36565  hashnexinj  42220  supxrgere  45431  supxrgelem  45435  supxrge  45436  iccdifprioo  45615  icccncfext  45984  stirlinglem5  46175  sge0repnf  46483  sge0split  46506  nnfoctbdjlem  46552  nabctnabc  47030