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

Theorem calemes 2775
 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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
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.maj . 2 𝑥(𝜑𝜓)
2 calemes.min . . 3 𝑥(𝜓 → ¬ 𝜒)
3 con2 137 . . . 4 ((𝜓 → ¬ 𝜒) → (𝜒 → ¬ 𝜓))
43alimi 1813 . . 3 (∀𝑥(𝜓 → ¬ 𝜒) → ∀𝑥(𝜒 → ¬ 𝜓))
52, 4ax-mp 5 . 2 𝑥(𝜒 → ¬ 𝜓)
61, 5camestres 2761 1 𝑥(𝜒 → ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400 This theorem is referenced by:  calemos  2778
 Copyright terms: Public domain W3C validator