ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl9 GIF version

Theorem syl9 72
Description: A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
Hypotheses
Ref Expression
syl9.1 (𝜑 → (𝜓𝜒))
syl9.2 (𝜃 → (𝜒𝜏))
Assertion
Ref Expression
syl9 (𝜑 → (𝜃 → (𝜓𝜏)))

Proof of Theorem syl9
StepHypRef Expression
1 syl9.1 . 2 (𝜑 → (𝜓𝜒))
2 syl9.2 . . 3 (𝜃 → (𝜒𝜏))
32a1i 9 . 2 (𝜑 → (𝜃 → (𝜒𝜏)))
41, 3syl5d 68 1 (𝜑 → (𝜃 → (𝜓𝜏)))
Colors of variables: wff set class
Syntax hints:  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7
This theorem is referenced by:  syl9r  73  com23  78  sylan9  406  pm4.79dc  873  pclem6  1337  bilukdc  1359  sbequi  1795  reuss2  3326  reupick  3330  elres  4825  funimass4  5440  fliftfun  5665  elabgf2  12914  bj-rspgt  12920
  Copyright terms: Public domain W3C validator