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 31004. 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  830  ecase2d  1047  efald  1590  necon1ai  2982  supgtoreq  9463  konigthlem  10653  indpi  10992  sqrmo  15418  2sqcoprm  27762  axtgupdim2  28933  ncoltgdim2  29028  ex-natded5.13  31016  bnj1204  35642  knoppndvlem10  37387  hashnexinj  43179  supxrgere  46345  supxrgelem  46349  supxrge  46350  iccdifprioo  46528  icccncfext  46897  stirlinglem5  47088  sge0repnf  47396  sge0split  47419  nnfoctbdjlem  47465  nabctnabc  48001