ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3syld GIF version

Theorem 3syld 57
Description: Triple syllogism deduction. (Contributed by Jeff Hankins, 4-Aug-2009.)
Hypotheses
Ref Expression
3syld.1 (𝜑 → (𝜓𝜒))
3syld.2 (𝜑 → (𝜒𝜃))
3syld.3 (𝜑 → (𝜃𝜏))
Assertion
Ref Expression
3syld (𝜑 → (𝜓𝜏))

Proof of Theorem 3syld
StepHypRef Expression
1 3syld.1 . . 3 (𝜑 → (𝜓𝜒))
2 3syld.2 . . 3 (𝜑 → (𝜒𝜃))
31, 2syld 45 . 2 (𝜑 → (𝜓𝜃))
4 3syld.3 . 2 (𝜑 → (𝜃𝜏))
53, 4syld 45 1 (𝜑 → (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:  xpfi  6931  fodjumkvlemres  7159  enmkvlem  7161  apreap  8546  msqge0  8575  cju  8920  facavg  10728  mulcn2  11322  coprm  12146  rpexp  12155  cnpnei  13758  lgseisenlem2  14490  ismkvnnlem  14839
  Copyright terms: Public domain W3C validator