ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl3an3 GIF version

Theorem syl3an3 1209
Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.)
Hypotheses
Ref Expression
syl3an3.1 (𝜑𝜃)
syl3an3.2 ((𝜓𝜒𝜃) → 𝜏)
Assertion
Ref Expression
syl3an3 ((𝜓𝜒𝜑) → 𝜏)

Proof of Theorem syl3an3
StepHypRef Expression
1 syl3an3.1 . . 3 (𝜑𝜃)
2 syl3an3.2 . . . 4 ((𝜓𝜒𝜃) → 𝜏)
323exp 1142 . . 3 (𝜓 → (𝜒 → (𝜃𝜏)))
41, 3syl7 68 . 2 (𝜓 → (𝜒 → (𝜑𝜏)))
543imp 1137 1 ((𝜓𝜒𝜑) → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-3an 926
This theorem is referenced by:  syl3an3b  1212  syl3an3br  1215  vtoclgft  2669  ovmpt2x  5755  ovmpt2ga  5756  nnanq0  6996  apreim  8056  divassap  8131  ltmul2  8289  elfzo  9525  fzodcel  9528  subcn2  10664  mulcn2  10665  ndvdsp1  11014  gcddiv  11090  lcmneg  11138
  Copyright terms: Public domain W3C validator