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Metamath Proof Explorer

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	$⊢ \forall x (φ \to ψ)$
	calemes.min	$⊢ \forall x (ψ \to \neg χ)$
Assertion	calemes	$⊢ \forall x (χ \to \neg φ)$

Proof

Step	Hyp	Ref	Expression
1		calemes.maj	$⊢ \forall x (φ \to ψ)$
2		calemes.min	$⊢ \forall x (ψ \to \neg χ)$
3		con2	$⊢ (ψ \to \neg χ) \to (χ \to \neg ψ)$
4	3	alimi	$⊢ \forall x (ψ \to \neg χ) \to \forall x (χ \to \neg ψ)$
5	2 4	ax-mp	$⊢ \forall x (χ \to \neg ψ)$
6	1 5	camestres	$⊢ \forall x (χ \to \neg φ)$