MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl6mpi Structured version   Visualization version   GIF version

Theorem syl6mpi 67
Description: A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)
Hypotheses
Ref Expression
syl6mpi.1 (𝜑 → (𝜓𝜒))
syl6mpi.2 𝜃
syl6mpi.3 (𝜒 → (𝜃𝜏))
Assertion
Ref Expression
syl6mpi (𝜑 → (𝜓𝜏))

Proof of Theorem syl6mpi
StepHypRef Expression
1 syl6mpi.1 . 2 (𝜑 → (𝜓𝜒))
2 syl6mpi.2 . . 3 𝜃
3 syl6mpi.3 . . 3 (𝜒 → (𝜃𝜏))
42, 3mpi 20 . 2 (𝜒𝜏)
51, 4syl6 35 1 (𝜑 → (𝜓𝜏))
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:  19.8a  2050  suceloni  6998  bndrank  8689  ac10ct  8842  1re  10024  uspgrn2crct  26681  tratrb  38566  ee20an  38776
  Copyright terms: Public domain W3C validator