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| Description: Double negation introduction. Converse of notnotr 130 and one implication of notnotb 315. Theorem *2.12 of [WhiteheadRussell] p. 101. This was the sixth axiom of Frege, specifically Proposition 41 of [Frege1879] p. 47. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.) | 
| Ref | Expression | 
|---|---|
| notnot | ⊢ (𝜑 → ¬ ¬ 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (¬ 𝜑 → ¬ 𝜑) | |
| 2 | 1 | con2i 139 | 1 ⊢ (𝜑 → ¬ ¬ 𝜑) | 
| 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: notnoti 143 notnotd 144 con1d 145 notnotb 315 pm2.13 897 biortn 937 necon2ad 2954 necon4ad 2958 necon4ai 2971 eueq2 3715 ifnot 4577 spthcycl 35135 knoppndvlem10 36523 wl-orel12 37513 cnfn1dd 38100 cnfn2dd 38101 axfrege41 43862 vk15.4j 44553 zfregs2VD 44866 vk15.4jVD 44939 con3ALTVD 44941 stoweidlem39 46059 | 
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