Theorem bocardo

Description: "Bocardo", one of the syllogisms of Aristotelian logic. Some ph is not ps , and all ph is ch , therefore some ch is not ps . Instance of disamis . In Aristotelian notation, OAO-3: MoP and MaS therefore SoP. For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". (Contributed by David A. Wheeler, 28-Aug-2016)

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

Proof

Step	Hyp	Ref	Expression
1		bocardo.maj	$$⊢\exists x\left(\phi \wedge \neg \psi \right)$$
2		bocardo.min	$$⊢\forall x\left(\phi \to \chi \right)$$
3	1 2	disamis	$$⊢\exists x\left(\chi \wedge \neg \psi \right)$$