Theorem felapton

Description: "Felapton", one of the syllogisms of Aristotelian logic. No ph is ps , all ph is ch , and some ph exist, therefore some ch is not ps . Instance of darapti . In Aristotelian notation, EAO-3: MeP and MaS therefore SoP. For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016)

	Ref	Expression
Hypotheses	felapton.maj	$$⊢\forall x\left(\phi \to \neg \psi \right)$$
	felapton.min	$$⊢\forall x\left(\phi \to \chi \right)$$
	felapton.e	$$⊢\exists x\phi$$
Assertion	felapton	$$⊢\exists x\left(\chi \wedge \neg \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		felapton.maj	$$⊢\forall x\left(\phi \to \neg \psi \right)$$
2		felapton.min	$$⊢\forall x\left(\phi \to \chi \right)$$
3		felapton.e	$$⊢\exists x\phi$$
4	1 2 3	darapti	$$⊢\exists x\left(\chi \wedge \neg \psi \right)$$