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Theorem cesaro 2311
 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 . . . . . 6 x(φ → ¬ ψ)
32spi 1753 . . . . 5 (φ → ¬ ψ)
4 cesaro.min . . . . . 6 x(χψ)
54spi 1753 . . . . 5 (χψ)
63, 5nsyl3 111 . . . 4 (χ → ¬ φ)
76ancli 534 . . 3 (χ → (χ ¬ φ))
87eximi 1576 . 2 (xχx(χ ¬ φ))
91, 8ax-mp 8 1 x(χ ¬ φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by: (None)
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