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

Theorem camestres 2673
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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
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.maj . . 3 𝑥(𝜑𝜓)
2 con3 156 . . . 4 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
32alimi 1819 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥𝜓 → ¬ 𝜑))
41, 3ax-mp 5 . 2 𝑥𝜓 → ¬ 𝜑)
5 camestres.min . 2 𝑥(𝜒 → ¬ 𝜓)
64, 5celarent 2664 1 𝑥(𝜒 → ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817
This theorem depends on definitions:  df-bi 210  df-an 400
This theorem is referenced by:  camestros  2679  calemes  2687
  Copyright terms: Public domain W3C validator