Theorem camestros

Description: "Camestros", one of the syllogisms of Aristotelian logic. All ph is ps , no ch is ps , and ch exist, therefore some ch is not ph . In Aristotelian notation, AEO-2: PaM and SeM therefore SoP. For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		camestros.maj	$$⊢\forall x\left(\phi \to \psi \right)$$
2		camestros.min	$$⊢\forall x\left(\chi \to \neg \psi \right)$$
3		camestros.e	$$⊢\exists x\chi$$
4	1 2	camestres	$$⊢\forall x\left(\chi \to \neg \phi \right)$$
5	3 4	barbarilem	$$⊢\exists x\left(\chi \wedge \neg \phi \right)$$