Theorem imim1 84

Description: A closed form of syllogism (see syl 18). Theorem *2.06 of [WhiteheadRussell] p. 100. Its associated inference is imim1i 64. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.)

Assertion

Ref	Expression
imim1	⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

Proof of Theorem imim1

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	imim1d 83	1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  pm2.83  85  peirceroll  86  imim12  106  looinv  206  pm3.33  777  a2and  859  impsingle  1660  tarski-bernays-ax2  1673  tbw-ax1  1733  moim  2574  sstr2  3945  ssralv  4007  mndind  18924  tb-ax1  36951  bj-imim21  37196  bj-imim11  37198  bj-alsyl  37271  bj-spimenfa  37304  al2imVD  45628  syl5impVD  45629  hbimpgVD  45670  hbalgVD  45671  ax6e2ndeqVD  45675  2sb5ndVD  45676