Metamath Proof Explorer


Theorem celarent

Description: "Celarent", one of the syllogisms of Aristotelian logic. No ph is ps , and all ch is ph , therefore no ch is ps . Instance of barbara . 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)

Ref Expression
Hypotheses celarent.maj 𝑥 ( 𝜑 → ¬ 𝜓 )
celarent.min 𝑥 ( 𝜒𝜑 )
Assertion celarent 𝑥 ( 𝜒 → ¬ 𝜓 )

Proof

Step Hyp Ref Expression
1 celarent.maj 𝑥 ( 𝜑 → ¬ 𝜓 )
2 celarent.min 𝑥 ( 𝜒𝜑 )
3 1 2 barbara 𝑥 ( 𝜒 → ¬ 𝜓 )