MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imim1 Structured version   Visualization version   GIF version

Theorem imim1 83
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.)
Assertion
Ref Expression
imim1 ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Proof of Theorem imim1
StepHypRef Expression
1 id 22 . 2 ((𝜑𝜓) → (𝜑𝜓))
21imim1d 82 1 ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
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  looinv  194  pm3.33  608  tbw-ax1  1665  19.38a  1807  moim  2548  mrcmndind  17413  tb-ax1  32503  bj-imim21  32664  al2imVD  39412  syl5impVD  39413  hbimpgVD  39454  hbalgVD  39455  ax6e2ndeqVD  39459  2sb5ndVD  39460
  Copyright terms: Public domain W3C validator