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Metamath Proof Explorer

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

Proof

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