Theorem imim1 81

Description: A closed form of syllogism (see syl 17). Theorem *2.06 of [WhiteheadRussell] p. 100. Its associated inference is imim1i 61. (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 80	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  82  peirceroll  83  looinv  193  pm3.33  607  tbw-ax1  1616  moim  2507  mrcmndind  17189  tb-ax1  31548  bj-imim21  31709  al2imVD  38120  syl5impVD  38121  hbimpgVD  38162  hbalgVD  38163  ax6e2ndeqVD  38167  2sb5ndVD  38168