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 30482. 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  818  ecase2d  1032  efald  1563  necon1ai  2960  supgtoreq  9378  konigthlem  10483  indpi  10822  sqrmo  15178  2sqcoprm  27406  axtgupdim2  28547  ncoltgdim2  28641  ex-natded5.13  30494  bnj1204  35170  knoppndvlem10  36723  hashnexinj  42450  supxrgere  45645  supxrgelem  45649  supxrge  45650  iccdifprioo  45829  icccncfext  46198  stirlinglem5  46389  sge0repnf  46697  sge0split  46720  nnfoctbdjlem  46766  nabctnabc  47244