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Theorem celaront 2717
Description: "Celaront", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
celaront.maj 𝑥(𝜑 → ¬ 𝜓)
celaront.min 𝑥(𝜒𝜑)
celaront.e 𝑥𝜒
Assertion
Ref Expression
celaront 𝑥(𝜒 ∧ ¬ 𝜓)

Proof of Theorem celaront
StepHypRef Expression
1 celaront.maj . 2 𝑥(𝜑 → ¬ 𝜓)
2 celaront.min . 2 𝑥(𝜒𝜑)
3 celaront.e . 2 𝑥𝜒
41, 2, 3barbari 2716 1 𝑥(𝜒 ∧ ¬ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by: (None)
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