Theorem exintr

Description: Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993) (Proof shortened by BJ, 16-Sep-2022)

	Ref	Expression
Assertion	exintr	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \exists x\left(\phi \wedge \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ancl	$$⊢\left(\phi \to \psi \right)\to \left(\phi \to \left(\phi \wedge \psi \right)\right)$$
2	1	aleximi	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \exists x\left(\phi \wedge \psi \right)\right)$$