Theorem datisi

Description: "Datisi", one of the syllogisms of Aristotelian logic. All ph is ps , and some ph is ch , therefore some ch is ps . In Aristotelian notation, AII-3: MaP and MiS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		datisi.maj	$$⊢\forall x\left(\phi \to \psi \right)$$
2		datisi.min	$$⊢\exists x\left(\phi \wedge \chi \right)$$
3		exancom	$$⊢\exists x\left(\phi \wedge \chi \right)\leftrightarrow \exists x\left(\chi \wedge \phi \right)$$
4	2 3	mpbi	$$⊢\exists x\left(\chi \wedge \phi \right)$$
5	1 4	darii	$$⊢\exists x\left(\chi \wedge \psi \right)$$