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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	$⊢ \forall x (φ \to \neg ψ)$
	celarent.min	$⊢ \forall x (χ \to φ)$
Assertion	celarent	$⊢ \forall x (χ \to \neg ψ)$

Proof

Step	Hyp	Ref	Expression
1		celarent.maj	$⊢ \forall x (φ \to \neg ψ)$
2		celarent.min	$⊢ \forall x (χ \to φ)$
3	1 2	barbara	$⊢ \forall x (χ \to \neg ψ)$