Theorem cesaro

Description: "Cesaro", one of the syllogisms of Aristotelian logic. No ph is ps , all ch is ps , and ch exist, therefore some ch is not ph . In Aristotelian notation, EAO-2: PeM and SaM therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

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