ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl2and GIF version

Theorem syl2and 293
Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
Hypotheses
Ref Expression
syl2and.1 (𝜑 → (𝜓𝜒))
syl2and.2 (𝜑 → (𝜃𝜏))
syl2and.3 (𝜑 → ((𝜒𝜏) → 𝜂))
Assertion
Ref Expression
syl2and (𝜑 → ((𝜓𝜃) → 𝜂))

Proof of Theorem syl2and
StepHypRef Expression
1 syl2and.1 . 2 (𝜑 → (𝜓𝜒))
2 syl2and.2 . . 3 (𝜑 → (𝜃𝜏))
3 syl2and.3 . . 3 (𝜑 → ((𝜒𝜏) → 𝜂))
42, 3sylan2d 292 . 2 (𝜑 → ((𝜒𝜃) → 𝜂))
51, 4syland 291 1 (𝜑 → ((𝜓𝜃) → 𝜂))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  anim12d  333  recexprlem1ssl  7566  recexprlem1ssu  7567  xle2add  9807  fzen  9969  bezoutlembi  11927  rpmulgcd2  12016  pcqmul  12224
  Copyright terms: Public domain W3C validator