![]() |
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 202 pm3.33 763 a2and 843 impsingle 1622 tarski-bernays-ax2 1635 tbw-ax1 1695 moim 2533 sstr2 3985 ssralv 4047 mndind 18813 tb-ax1 36108 bj-imim21 36267 bj-cbvalimt 36356 bj-cbveximt 36357 al2imVD 44575 syl5impVD 44576 hbimpgVD 44617 hbalgVD 44618 ax6e2ndeqVD 44622 2sb5ndVD 44623 |
Copyright terms: Public domain | W3C validator |