![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > imim1 | Structured version Visualization version GIF version |
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 pm3.33 765 a2and 845 impsingle 1624 tarski-bernays-ax2 1637 tbw-ax1 1697 moim 2542 sstr2 4002 ssralv 4064 mndind 18854 tb-ax1 36366 bj-imim21 36534 bj-cbvalimt 36622 bj-cbveximt 36623 al2imVD 44860 syl5impVD 44861 hbimpgVD 44902 hbalgVD 44903 ax6e2ndeqVD 44907 2sb5ndVD 44908 |
Copyright terms: Public domain | W3C validator |