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	$$⊢\forall x\left(\phi \to \neg \psi \right)$$
	celarent.min	$$⊢\forall x\left(\chi \to \phi \right)$$
Assertion	celarent	$$⊢\forall x\left(\chi \to \neg \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		celarent.maj	$$⊢\forall x\left(\phi \to \neg \psi \right)$$
2		celarent.min	$$⊢\forall x\left(\chi \to \phi \right)$$
3	1 2	barbara	$$⊢\forall x\left(\chi \to \neg \psi \right)$$