|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > notnotrd | Structured version Visualization version GIF version | ||
| 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 30422. 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.) | 
| Ref | Expression | 
|---|---|
| notnotrd.1 | ⊢ (𝜑 → ¬ ¬ 𝜓) | 
| Ref | Expression | 
|---|---|
| notnotrd | ⊢ (𝜑 → 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | notnotrd.1 | . 2 ⊢ (𝜑 → ¬ ¬ 𝜓) | |
| 2 | notnotr 130 | . 2 ⊢ (¬ ¬ 𝜓 → 𝜓) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝜓) | 
| Colors of variables: wff setvar class | 
| 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 1561 necon1ai 2968 supgtoreq 9510 konigthlem 10608 indpi 10947 sqrmo 15290 2sqcoprm 27479 axtgupdim2 28479 ncoltgdim2 28573 ex-natded5.13 30434 bnj1204 35026 knoppndvlem10 36522 hashnexinj 42129 supxrgere 45344 supxrgelem 45348 supxrge 45349 iccdifprioo 45529 icccncfext 45902 stirlinglem5 46093 sge0repnf 46401 sge0split 46424 nnfoctbdjlem 46470 nabctnabc 46943 | 
| Copyright terms: Public domain | W3C validator |