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)))

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  85  peirceroll  86  imim12  106  looinv  206  pm3.33  776  a2and  858  impsingle  1657  tarski-bernays-ax2  1670  tbw-ax1  1730  moim  2572  sstr2  3944  ssralv  4006  mndind  18882  tb-ax1  36894  bj-imim21  37139  bj-imim11  37141  bj-alsyl  37214  bj-spimenfa  37247  al2imVD  45570  syl5impVD  45571  hbimpgVD  45612  hbalgVD  45613  ax6e2ndeqVD  45617  2sb5ndVD  45618