Theorem festino

Description: "Festino", one of the syllogisms of Aristotelian logic. No ph is ps , and some ch is ps , therefore some ch is not ph . In Aristotelian notation, EIO-2: PeM and SiM therefore SoP. (Contributed by David A. Wheeler, 25-Nov-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		festino.maj	$$⊢\forall x\left(\phi \to \neg \psi \right)$$
2		festino.min	$$⊢\exists x\left(\chi \wedge \psi \right)$$
3		con2	$$⊢\left(\phi \to \neg \psi \right)\to \left(\psi \to \neg \phi \right)$$
4	3	anim2d	$$⊢\left(\phi \to \neg \psi \right)\to \left(\left(\chi \wedge \psi \right)\to \left(\chi \wedge \neg \phi \right)\right)$$
5	4	alimi	$$⊢\forall x\left(\phi \to \neg \psi \right)\to \forall x\left(\left(\chi \wedge \psi \right)\to \left(\chi \wedge \neg \phi \right)\right)$$
6	1 5	ax-mp	$$⊢\forall x\left(\left(\chi \wedge \psi \right)\to \left(\chi \wedge \neg \phi \right)\right)$$
7		exim	$$⊢\forall x\left(\left(\chi \wedge \psi \right)\to \left(\chi \wedge \neg \phi \right)\right)\to \left(\exists x\left(\chi \wedge \psi \right)\to \exists x\left(\chi \wedge \neg \phi \right)\right)$$
8	6 2 7	mp2	$$⊢\exists x\left(\chi \wedge \neg \phi \right)$$