Theorem cesare

Description: "Cesare", one of the syllogisms of Aristotelian logic. No ph is ps , and all ch is ps , therefore no ch is ph . In Aristotelian notation, EAE-2: PeM and SaM therefore SeP. Related to celarent . (Contributed by David A. Wheeler, 27-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		cesare.maj	$$⊢\forall x\left(\phi \to \neg \psi \right)$$
2		cesare.min	$$⊢\forall x\left(\chi \to \psi \right)$$
3		con2	$$⊢\left(\phi \to \neg \psi \right)\to \left(\psi \to \neg \phi \right)$$
4	3	alimi	$$⊢\forall x\left(\phi \to \neg \psi \right)\to \forall x\left(\psi \to \neg \phi \right)$$
5	1 4	ax-mp	$$⊢\forall x\left(\psi \to \neg \phi \right)$$
6	5 2	celarent	$$⊢\forall x\left(\chi \to \neg \phi \right)$$