MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imim2 Structured version   Visualization version   GIF version

Theorem imim2 58
Description: A closed form of syllogism (see syl 17). Theorem *2.05 of [WhiteheadRussell] p. 100. Its associated inference is imim2i 16. Its associated deduction is imim2d 57. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 6-Sep-2012.)
Assertion
Ref Expression
imim2 ((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Proof of Theorem imim2
StepHypRef Expression
1 id 22 . 2 ((𝜑𝜓) → (𝜑𝜓))
21imim2d 57 1 ((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))
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:  syldd  72  peirceroll  85  imim12  105  pm3.34  609  19.38b  1765  bj-ssbim  32263  19.41rgVD  38621
  Copyright terms: Public domain W3C validator