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

Theorem sylcom 28
Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)
Hypotheses
Ref Expression
sylcom.1 (𝜑 → (𝜓𝜒))
sylcom.2 (𝜓 → (𝜒𝜃))
Assertion
Ref Expression
sylcom (𝜑 → (𝜓𝜃))

Proof of Theorem sylcom
StepHypRef Expression
1 sylcom.1 . 2 (𝜑 → (𝜓𝜒))
2 sylcom.2 . . 3 (𝜓 → (𝜒𝜃))
32a2i 11 . 2 ((𝜓𝜒) → (𝜓𝜃))
41, 3syl 14 1 (𝜑 → (𝜓𝜃))
Colors of variables: wff set class
Syntax hints:  wi 4
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7
This theorem is referenced by:  syl5com  29  syl6  33  syli  37  mpbidi  149  con4biddc  792  jaddc  799  con1biddc  808  necon4addc  2325  necon4bddc  2326  necon4ddc  2327  necon1addc  2331  necon1bddc  2332  dmcosseq  4704  iss  4758  funopg  5048  snon0  6643
  Copyright terms: Public domain W3C validator