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Theorem felapton 2715
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. Instance of darapti 2713. 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.)
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.maj . 2 𝑥(𝜑 → ¬ 𝜓)
2 felapton.min . 2 𝑥(𝜑𝜒)
3 felapton.e . 2 𝑥𝜑
41, 2, 3darapti 2713 1 𝑥(𝜒 ∧ ¬ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1650  wex 1874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875
This theorem is referenced by:  fesapo  2724
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