Theorem calemes

Description: "Calemes", one of the syllogisms of Aristotelian logic. All ph is ps , and no ps is ch , therefore no ch is ph . In Aristotelian notation, AEE-4: PaM and MeS therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

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