ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl3anl2 GIF version

Theorem syl3anl2 1195
Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.)
Hypotheses
Ref Expression
syl3anl2.1 (𝜑𝜒)
syl3anl2.2 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
Assertion
Ref Expression
syl3anl2 (((𝜓𝜑𝜃) ∧ 𝜏) → 𝜂)

Proof of Theorem syl3anl2
StepHypRef Expression
1 syl3anl2.1 . . 3 (𝜑𝜒)
2 syl3anl2.2 . . . 4 (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)
32ex 112 . . 3 ((𝜓𝜒𝜃) → (𝜏𝜂))
41, 3syl3an2 1180 . 2 ((𝜓𝜑𝜃) → (𝜏𝜂))
54imp 119 1 (((𝜓𝜑𝜃) ∧ 𝜏) → 𝜂)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114  df-3an 898
This theorem is referenced by:  syl3anr2  1199
  Copyright terms: Public domain W3C validator