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

Theorem syl3anl1 1409
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 1149 . 2 ((𝜑𝜒𝜃) → (𝜓𝜒𝜃))
3 syl3anl1.2 . 2 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
42, 3sylan 579 1 (((𝜑𝜒𝜃) ∧ 𝜏) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084
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 206  df-an 396  df-3an 1086
This theorem is referenced by:  dif1enlem  9158  suprzcl  12646  latjcom  18412  latmcom  18428  ring1zr  20627  lgsdinn0  27233  revpfxsfxrev  34634  crngohomfo  37387  dalem53  39109
  Copyright terms: Public domain W3C validator