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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  imim12  105  looinv  206  pm3.33  764  a2and  842  impsingle  1629  tarski-bernays-ax2  1642  tbw-ax1  1702  moim  2602  mndind  18001  tb-ax1  33907  bj-imim21  34066  bj-cbvalimt  34152  bj-cbveximt  34153  al2imVD  41655  syl5impVD  41656  hbimpgVD  41697  hbalgVD  41698  ax6e2ndeqVD  41702  2sb5ndVD  41703
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