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 30765. 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)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  condan  829  ecase2d  1046  efald  1590  necon1ai  2984  supgtoreq  9429  konigthlem  10559  indpi  10898  sqrmo  15309  2sqcoprm  27610  axtgupdim2  28751  ncoltgdim2  28845  ex-natded5.13  30777  bnj1204  35409  knoppndvlem10  37138  hashnexinj  42923  supxrgere  46077  supxrgelem  46081  supxrge  46082  iccdifprioo  46260  icccncfext  46629  stirlinglem5  46820  sge0repnf  47128  sge0split  47151  nnfoctbdjlem  47197  nabctnabc  47696