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Theorem cesaro 2557
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 2040 . . . 4 (𝜑 → ¬ 𝜓)
4 cesaro.min . . . . 5 𝑥(𝜒𝜓)
54spi 2040 . . . 4 (𝜒𝜓)
63, 5nsyl3 131 . . 3 (𝜒 → ¬ 𝜑)
76ancli 571 . 2 (𝜒 → (𝜒 ∧ ¬ 𝜑))
81, 7eximii 1753 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by: (None)
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