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

Theorem sylani 607
Description: A syllogism inference. (Contributed by NM, 2-May-1996.)
Hypotheses
Ref Expression
sylani.1 (𝜑𝜒)
sylani.2 (𝜓 → ((𝜒𝜃) → 𝜏))
Assertion
Ref Expression
sylani (𝜓 → ((𝜑𝜃) → 𝜏))

Proof of Theorem sylani
StepHypRef Expression
1 sylani.1 . . 3 (𝜑𝜒)
21a1i 11 . 2 (𝜓 → (𝜑𝜒))
3 sylani.2 . 2 (𝜓 → ((𝜒𝜃) → 𝜏))
42, 3syland 606 1 (𝜓 → ((𝜑𝜃) → 𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 210  df-an 400
This theorem is referenced by:  syl2ani  610  inf3lem2  9158  zorn2lem5  9993  uzwo  12386  supxrun  12785  lcmdvds  16042  cramer0  21434  csmdsymi  30261  matunitlindflem2  35386  pmapjoin  37478
  Copyright terms: Public domain W3C validator