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 30473. 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  2959  supgtoreq  9384  konigthlem  10491  indpi  10830  sqrmo  15213  2sqcoprm  27398  axtgupdim2  28539  ncoltgdim2  28633  ex-natded5.13  30485  bnj1204  35154  knoppndvlem10  36781  hashnexinj  42567  supxrgere  45763  supxrgelem  45767  supxrge  45768  iccdifprioo  45946  icccncfext  46315  stirlinglem5  46506  sge0repnf  46814  sge0split  46837  nnfoctbdjlem  46883  nabctnabc  47379