ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl3anr2 GIF version

Theorem syl3anr2 1301
Description: A syllogism inference. (Contributed by NM, 1-Aug-2007.)
Hypotheses
Ref Expression
syl3anr2.1 (𝜑𝜃)
syl3anr2.2 ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)
Assertion
Ref Expression
syl3anr2 ((𝜒 ∧ (𝜓𝜑𝜏)) → 𝜂)

Proof of Theorem syl3anr2
StepHypRef Expression
1 syl3anr2.1 . . 3 (𝜑𝜃)
2 syl3anr2.2 . . . 4 ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)
32ancoms 268 . . 3 (((𝜓𝜃𝜏) ∧ 𝜒) → 𝜂)
41, 3syl3anl2 1297 . 2 (((𝜓𝜑𝜏) ∧ 𝜒) → 𝜂)
54ancoms 268 1 ((𝜒 ∧ (𝜓𝜑𝜏)) → 𝜂)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108
This theorem depends on definitions:  df-bi 117  df-3an 981
This theorem is referenced by:  mulgsubdir  13052
  Copyright terms: Public domain W3C validator