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Metamath Proof Explorer

Theorem felapton

Description: "Felapton", one of the syllogisms of Aristotelian logic. No ph is ps , all ph is ch , and some ph exist, therefore some ch is not ps . Instance of darapti . In Aristotelian notation, EAO-3: MeP and MaS therefore SoP. For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016)

	Ref	Expression
Hypotheses	felapton.maj	$⊢ \forall x (φ \to \neg ψ)$
	felapton.min	$⊢ \forall x (φ \to χ)$
	felapton.e	$⊢ \exists x φ$
Assertion	felapton	$⊢ \exists x (χ \land \neg ψ)$

Proof

Step	Hyp	Ref	Expression
1		felapton.maj	$⊢ \forall x (φ \to \neg ψ)$
2		felapton.min	$⊢ \forall x (φ \to χ)$
3		felapton.e	$⊢ \exists x φ$
4	1 2 3	darapti	$⊢ \exists x (χ \land \neg ψ)$