Theorem disamis

Description: "Disamis", one of the syllogisms of Aristotelian logic. Some ph is ps , and all ph is ch , therefore some ch is ps . In Aristotelian notation, IAI-3: MiP and MaS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

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