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Theorem celarent 2721
Description: "Celarent", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore no 𝜒 is 𝜓. Instance of barbara 2720. In Aristotelian notation, EAE-1: MeP and SaM therefore SeP. For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
Hypotheses
Ref Expression
celarent.maj 𝑥(𝜑 → ¬ 𝜓)
celarent.min 𝑥(𝜒𝜑)
Assertion
Ref Expression
celarent 𝑥(𝜒 → ¬ 𝜓)

Proof of Theorem celarent
StepHypRef Expression
1 celarent.maj . 2 𝑥(𝜑 → ¬ 𝜓)
2 celarent.min . 2 𝑥(𝜒𝜑)
31, 2barbara 2720 1 𝑥(𝜒 → ¬ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905
This theorem depends on definitions:  df-bi 199  df-an 386
This theorem is referenced by:  cesare  2729  camestres  2731
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