Theorem camestres

Description: "Camestres", one of the syllogisms of Aristotelian logic. All ph is ps , and no ch is ps , therefore no ch is ph . In Aristotelian notation, AEE-2: PaM and SeM therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		camestres.maj	$$⊢\forall x\left(\phi \to \psi \right)$$
2		camestres.min	$$⊢\forall x\left(\chi \to \neg \psi \right)$$
3		con3	$$⊢\left(\phi \to \psi \right)\to \left(\neg \psi \to \neg \phi \right)$$
4	3	alimi	$$⊢\forall x\left(\phi \to \psi \right)\to \forall x\left(\neg \psi \to \neg \phi \right)$$
5	1 4	ax-mp	$$⊢\forall x\left(\neg \psi \to \neg \phi \right)$$
6	5 2	celarent	$$⊢\forall x\left(\chi \to \neg \phi \right)$$