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Theorem celaront 2045
 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 2044 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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