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	⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

Proof of Theorem imim1

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	imim1d 82	1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

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  205  pm3.33  774  a2and  856  impsingle  1647  tarski-bernays-ax2  1660  tbw-ax1  1720  moim  2571  sstr2  3943  ssralv  4005  mndind  18862  tb-ax1  36740  bj-imim21  36986  bj-imim11  36988  bj-alsyl  37061  bj-spimenfa  37094  al2imVD  45434  syl5impVD  45435  hbimpgVD  45476  hbalgVD  45477  ax6e2ndeqVD  45481  2sb5ndVD  45482