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 132 and notnotri 133. Double negation elimination rule. A translation of the natural deduction rule ¬ ¬ C , Γ⊢ ¬ ¬ 𝜓 ⇒ Γ⊢ 𝜓; see natded 28340. 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 132 | . 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 1029 efald 1563 necon1ai 2961 supgtoreq 9007 konigthlem 10068 indpi 10407 sqrmo 14701 2sqcoprm 26171 axtgupdim2 26417 ncoltgdim2 26511 ex-natded5.13 28352 bnj1204 32563 knoppndvlem10 34339 supxrgere 42410 supxrgelem 42414 supxrge 42415 iccdifprioo 42594 icccncfext 42970 stirlinglem5 43161 sge0repnf 43466 sge0split 43489 nnfoctbdjlem 43535 nabctnabc 43965 |
Copyright terms: Public domain | W3C validator |