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

Theorem calemes 2013
Description: "Calemes", one of the syllogisms of Aristotelian logic. All φ is ψ, and no ψ is χ, therefore no χ is φ. (In Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
calemes.maj x(φψ)
calemes.min x(ψ → ¬ χ)
Assertion
Ref Expression
calemes x(χ → ¬ φ)

Proof of Theorem calemes
StepHypRef Expression
1 calemes.min . . . . 5 x(ψ → ¬ χ)
21spi 1426 . . . 4 (ψ → ¬ χ)
32con2i 557 . . 3 (χ → ¬ ψ)
4 calemes.maj . . . 4 x(φψ)
54spi 1426 . . 3 (φψ)
63, 5nsyl 558 . 2 (χ → ¬ φ)
76ax-gen 1335 1 x(χ → ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-in1 544  ax-in2 545  ax-gen 1335  ax-4 1397
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator