Theorem exan

Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 13-Jan-2018) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021) (Proof shortened by Wolf Lammen, 6-Nov-2022) Expand hypothesis. (Revised by Steven Nguyen, 19-Jun-2023)

	Ref	Expression
Hypotheses	exan.1	$$⊢\exists x\phi$$
	exan.2	$$⊢\psi$$
Assertion	exan	$$⊢\exists x\left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		exan.1	$$⊢\exists x\phi$$
2		exan.2	$$⊢\psi$$
3	2	jctr	$$⊢\phi \to \left(\phi \wedge \psi \right)$$
4	1 3	eximii	$$⊢\exists x\left(\phi \wedge \psi \right)$$