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Theorem felapton 2317
 Description: "Felapton", one of the syllogisms of Aristotelian logic. No φ is ψ, all φ is χ, and some φ exist, therefore some χ is not ψ. (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
felapton.maj x(φ → ¬ ψ)
felapton.min x(φχ)
felapton.e xφ
Assertion
Ref Expression
felapton x(χ ¬ ψ)

Proof of Theorem felapton
StepHypRef Expression
1 felapton.e . 2 xφ
2 felapton.min . . . . 5 x(φχ)
32spi 1753 . . . 4 (φχ)
4 felapton.maj . . . . 5 x(φ → ¬ ψ)
54spi 1753 . . . 4 (φ → ¬ ψ)
63, 5jca 518 . . 3 (φ → (χ ¬ ψ))
76eximi 1576 . 2 (xφx(χ ¬ ψ))
81, 7ax-mp 8 1 x(χ ¬ ψ)
 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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