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 30495. 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  824  ecase2d  1038  efald  1569  necon1ai  2963  supgtoreq  9378  konigthlem  10486  indpi  10825  sqrmo  15208  2sqcoprm  27420  axtgupdim2  28561  ncoltgdim2  28655  ex-natded5.13  30507  bnj1204  35209  knoppndvlem10  36842  hashnexinj  42628  supxrgere  45792  supxrgelem  45796  supxrge  45797  iccdifprioo  45975  icccncfext  46344  stirlinglem5  46535  sge0repnf  46843  sge0split  46866  nnfoctbdjlem  46912  nabctnabc  47408