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Theorem felapton 2057
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 . . . 4 𝑥(𝜑𝜒)
32spi 1470 . . 3 (𝜑𝜒)
4 felapton.maj . . . 4 𝑥(𝜑 → ¬ 𝜓)
54spi 1470 . . 3 (𝜑 → ¬ 𝜓)
63, 5jca 300 . 2 (𝜑 → (𝜒 ∧ ¬ 𝜓))
71, 6eximii 1534 1 𝑥(𝜒 ∧ ¬ 𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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