Theorem barbarilem

Description: Lemma for barbari and the other Aristotelian syllogisms with existential assumption. (Contributed by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		barbarilem.min	$$⊢\exists x\phi$$
2		barbarilem.maj	$$⊢\forall x\left(\phi \to \psi \right)$$
3		exintr	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \exists x\left(\phi \wedge \psi \right)\right)$$
4	2 1 3	mp2	$$⊢\exists x\left(\phi \wedge \psi \right)$$