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| Description: A closed form of syllogism (see syl 17). Theorem *2.06 of [WhiteheadRussell] p. 100. Its associated inference is imim1i 63. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.) | 
| Ref | Expression | 
|---|---|
| imim1 | ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
| 2 | 1 | imim1d 82 | 1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 | 
| This theorem is referenced by: pm2.83 84 peirceroll 85 imim12 105 looinv 203 pm3.33 764 a2and 845 impsingle 1626 tarski-bernays-ax2 1639 tbw-ax1 1699 moim 2543 sstr2 3989 ssralv 4051 mndind 18842 tb-ax1 36385 bj-imim21 36553 bj-cbvalimt 36641 bj-cbveximt 36642 al2imVD 44887 syl5impVD 44888 hbimpgVD 44929 hbalgVD 44930 ax6e2ndeqVD 44934 2sb5ndVD 44935 | 
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