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Mirrors > Home > MPE Home > Th. List > notnotrd | Structured version Visualization version GIF version |
Description: Deduction associated with notnotr 128 and notnotri 129. Double negation elimination rule. A translation of the natural deduction rule ¬ ¬ C , Γ⊢ ¬ ¬ 𝜓 ⇒ Γ⊢ 𝜓; see natded 27814. 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 128 | . 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 852 efald 1678 necon1ai 3026 supgtoreq 8651 konigthlem 9712 indpi 10051 sqrmo 14376 axtgupdim2 25790 ncoltgdim2 25884 ex-natded5.13 27826 2sqcoprm 30188 bnj1204 31622 knoppndvlem10 33039 supxrgere 40344 supxrgelem 40348 supxrge 40349 iccdifprioo 40536 icccncfext 40893 stirlinglem5 41087 sge0repnf 41392 sge0split 41415 nnfoctbdjlem 41461 nabctnabc 41890 |
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