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Theorem calemos 2776
Description: "Calemos", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓 (PaM), no 𝜓 is 𝜒 (MeS), and 𝜒 exist, therefore some 𝜒 is not 𝜑 (SoP). In Aristotelian notation, AEO-4: PaM and MeS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
Hypotheses
Ref Expression
calemos.maj 𝑥(𝜑𝜓)
calemos.min 𝑥(𝜓 → ¬ 𝜒)
calemos.e 𝑥𝜒
Assertion
Ref Expression
calemos 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem calemos
StepHypRef Expression
1 calemos.e . 2 𝑥𝜒
2 calemos.maj . . 3 𝑥(𝜑𝜓)
3 calemos.min . . 3 𝑥(𝜓 → ¬ 𝜒)
42, 3calemes 2773 . 2 𝑥(𝜒 → ¬ 𝜑)
51, 4barbarilem 2754 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1536  wex 1781
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  df-ex 1782
This theorem is referenced by: (None)
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