![]() |
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 pm3.33 764 a2and 844 impsingle 1630 tarski-bernays-ax2 1643 tbw-ax1 1703 moim 2539 mndind 18709 tb-ax1 35268 bj-imim21 35427 bj-cbvalimt 35516 bj-cbveximt 35517 al2imVD 43623 syl5impVD 43624 hbimpgVD 43665 hbalgVD 43666 ax6e2ndeqVD 43670 2sb5ndVD 43671 |
Copyright terms: Public domain | W3C validator |