MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl3anl1 Structured version   Visualization version   GIF version

Theorem syl3anl1 1534
Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.)
Hypotheses
Ref Expression
syl3anl1.1 (𝜑𝜓)
syl3anl1.2 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
Assertion
Ref Expression
syl3anl1 (((𝜑𝜒𝜃) ∧ 𝜏) → 𝜂)

Proof of Theorem syl3anl1
StepHypRef Expression
1 syl3anl1.1 . . 3 (𝜑𝜓)
213anim1i 1192 . 2 ((𝜑𝜒𝜃) → (𝜓𝜒𝜃))
3 syl3anl1.2 . 2 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
42, 3sylan 576 1 (((𝜑𝜒𝜃) ∧ 𝜏) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108
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 199  df-an 386  df-3an 1110
This theorem is referenced by:  suprzcl  11747  latjcom  17374  latmcom  17390  ring1zr  19598  lgsdinn0  25422  crngohomfo  34292  dalem53  35746
  Copyright terms: Public domain W3C validator