Theorem euanv

Description: Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023)

	Ref	Expression
Assertion	euanv	$$⊢\exists !x\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \exists !x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		euex	$$⊢\exists !x\left(\phi \wedge \psi \right)\to \exists x\left(\phi \wedge \psi \right)$$
2		simpl	$$⊢\left(\phi \wedge \psi \right)\to \phi$$
3	2	exlimiv	$$⊢\exists x\left(\phi \wedge \psi \right)\to \phi$$
4	1 3	syl	$$⊢\exists !x\left(\phi \wedge \psi \right)\to \phi$$
5		ibar	$$⊢\phi \to \left(\psi \leftrightarrow \left(\phi \wedge \psi \right)\right)$$
6	5	eubidv	$$⊢\phi \to \left(\exists !x\psi \leftrightarrow \exists !x\left(\phi \wedge \psi \right)\right)$$
7	6	biimprcd	$$⊢\exists !x\left(\phi \wedge \psi \right)\to \left(\phi \to \exists !x\psi \right)$$
8	4 7	jcai	$$⊢\exists !x\left(\phi \wedge \psi \right)\to \left(\phi \wedge \exists !x\psi \right)$$
9	6	biimpa	$$⊢\left(\phi \wedge \exists !x\psi \right)\to \exists !x\left(\phi \wedge \psi \right)$$
10	8 9	impbii	$$⊢\exists !x\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \exists !x\psi \right)$$