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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 28188. 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 817 efald 1559 necon1ai 3014 supgtoreq 8918 konigthlem 9979 indpi 10318 sqrmo 14603 2sqcoprm 26019 axtgupdim2 26265 ncoltgdim2 26359 ex-natded5.13 28200 bnj1204 32394 knoppndvlem10 33973 supxrgere 41965 supxrgelem 41969 supxrge 41970 iccdifprioo 42153 icccncfext 42529 stirlinglem5 42720 sge0repnf 43025 sge0split 43048 nnfoctbdjlem 43094 nabctnabc 43524 |
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