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Theorem camestros 2739
Description: "Camestros", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, no 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. In Aristotelian notation, AEO-2: PaM and SeM therefore SoP. For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
Hypotheses
Ref Expression
camestros.maj 𝑥(𝜑𝜓)
camestros.min 𝑥(𝜒 → ¬ 𝜓)
camestros.e 𝑥𝜒
Assertion
Ref Expression
camestros 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem camestros
StepHypRef Expression
1 camestros.e . 2 𝑥𝜒
2 camestros.maj . . 3 𝑥(𝜑𝜓)
3 camestros.min . . 3 𝑥(𝜒 → ¬ 𝜓)
42, 3camestres 2731 . 2 𝑥(𝜒 → ¬ 𝜑)
51, 4barbarilem 2725 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wal 1651  wex 1875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876
This theorem is referenced by: (None)
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