Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  camestres Structured version   Visualization version   GIF version

Theorem camestres 2719
 Description: "Camestres", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. (In Aristotelian notation, AEE-2: PaM and SeM therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
camestres.maj 𝑥(𝜑𝜓)
camestres.min 𝑥(𝜒 → ¬ 𝜓)
Assertion
Ref Expression
camestres 𝑥(𝜒 → ¬ 𝜑)

Proof of Theorem camestres
StepHypRef Expression
1 camestres.min . . . 4 𝑥(𝜒 → ¬ 𝜓)
21spi 2208 . . 3 (𝜒 → ¬ 𝜓)
3 camestres.maj . . . 4 𝑥(𝜑𝜓)
43spi 2208 . . 3 (𝜑𝜓)
52, 4nsyl 137 . 2 (𝜒 → ¬ 𝜑)
65ax-gen 1870 1 𝑥(𝜒 → ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203 This theorem depends on definitions:  df-bi 197  df-ex 1853 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator