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Theorem cesaro 2764
Description: "Cesaro", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. In Aristotelian notation, EAO-2: PeM and SaM therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
Hypotheses
Ref Expression
cesaro.maj 𝑥(𝜑 → ¬ 𝜓)
cesaro.min 𝑥(𝜒𝜓)
cesaro.e 𝑥𝜒
Assertion
Ref Expression
cesaro 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem cesaro
StepHypRef Expression
1 cesaro.e . 2 𝑥𝜒
2 cesaro.maj . . 3 𝑥(𝜑 → ¬ 𝜓)
3 cesaro.min . . 3 𝑥(𝜒𝜓)
42, 3cesare 2758 . 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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