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Theorem celaront 2139
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 2138 1 𝑥(𝜒 ∧ ¬ 𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wal 1361  wex 1502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-ial 1544
This theorem depends on definitions:  df-bi 117
This theorem is referenced by: (None)
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