Theorem baroco

Description: "Baroco", one of the syllogisms of Aristotelian logic. All ph is ps , and some ch is not ps , therefore some ch is not ph . In Aristotelian notation, AOO-2: PaM and SoM therefore SoP. For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative". (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		baroco.maj	$$⊢\forall x\left(\phi \to \psi \right)$$
2		baroco.min	$$⊢\exists x\left(\chi \wedge \neg \psi \right)$$
3		con3	$$⊢\left(\phi \to \psi \right)\to \left(\neg \psi \to \neg \phi \right)$$
4	3	anim2d	$$⊢\left(\phi \to \psi \right)\to \left(\left(\chi \wedge \neg \psi \right)\to \left(\chi \wedge \neg \phi \right)\right)$$
5	4	alimi	$$⊢\forall x\left(\phi \to \psi \right)\to \forall x\left(\left(\chi \wedge \neg \psi \right)\to \left(\chi \wedge \neg \phi \right)\right)$$
6	1 5	ax-mp	$$⊢\forall x\left(\left(\chi \wedge \neg \psi \right)\to \left(\chi \wedge \neg \phi \right)\right)$$
7		exim	$$⊢\forall x\left(\left(\chi \wedge \neg \psi \right)\to \left(\chi \wedge \neg \phi \right)\right)\to \left(\exists x\left(\chi \wedge \neg \psi \right)\to \exists x\left(\chi \wedge \neg \phi \right)\right)$$
8	6 2 7	mp2	$$⊢\exists x\left(\chi \wedge \neg \phi \right)$$