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Theorem cesaro 2024
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.) (Revised by David A. Wheeler, 2-Sep-2016.)
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 . . . . 5 𝑥(𝜑 → ¬ 𝜓)
32spi 1445 . . . 4 (𝜑 → ¬ 𝜓)
4 cesaro.min . . . . 5 𝑥(𝜒𝜓)
54spi 1445 . . . 4 (𝜒𝜓)
63, 5nsyl3 566 . . 3 (𝜒 → ¬ 𝜑)
76ancli 310 . 2 (𝜒 → (𝜒 ∧ ¬ 𝜑))
81, 7eximii 1509 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wal 1257  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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