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

Theorem cesare 2081
Description: "Cesare", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2076. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
Hypotheses
Ref Expression
cesare.maj 𝑥(𝜑 → ¬ 𝜓)
cesare.min 𝑥(𝜒𝜓)
Assertion
Ref Expression
cesare 𝑥(𝜒 → ¬ 𝜑)

Proof of Theorem cesare
StepHypRef Expression
1 cesare.maj . . . 4 𝑥(𝜑 → ¬ 𝜓)
21spi 1501 . . 3 (𝜑 → ¬ 𝜓)
3 cesare.min . . . 4 𝑥(𝜒𝜓)
43spi 1501 . . 3 (𝜒𝜓)
52, 4nsyl3 600 . 2 (𝜒 → ¬ 𝜑)
65ax-gen 1410 1 𝑥(𝜒 → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 588  ax-in2 589  ax-gen 1410  ax-4 1472
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator