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Metamath Proof Explorer

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

Proof

Step	Hyp	Ref	Expression
1		camestres.maj	$⊢ \forall x (φ \to ψ)$
2		camestres.min	$⊢ \forall x (χ \to \neg ψ)$
3		con3	$⊢ (φ \to ψ) \to (\neg ψ \to \neg φ)$
4	3	alimi	$⊢ \forall x (φ \to ψ) \to \forall x (\neg ψ \to \neg φ)$
5	1 4	ax-mp	$⊢ \forall x (\neg ψ \to \neg φ)$
6	5 2	celarent	$⊢ \forall x (χ \to \neg φ)$