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Theorem celaront 2669
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. Instance of barbari 2667. 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 2667. (Contributed by David A. Wheeler, 27-Aug-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 2667 1 𝑥(𝜒 ∧ ¬ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777
This theorem is referenced by: (None)
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