Theorem euorv

Description: Introduce a disjunct into a unique existential quantifier. Version of euor requiring disjoint variables, but fewer axioms. (Contributed by NM, 23-Mar-1995) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023)

	Ref	Expression
Assertion	euorv	$$⊢\left(\neg \phi \wedge \exists !x\psi \right)\to \exists !x\left(\phi \vee \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		biorf	$$⊢\neg \phi \to \left(\psi \leftrightarrow \left(\phi \vee \psi \right)\right)$$
2	1	eubidv	$$⊢\neg \phi \to \left(\exists !x\psi \leftrightarrow \exists !x\left(\phi \vee \psi \right)\right)$$
3	2	biimpa	$$⊢\left(\neg \phi \wedge \exists !x\psi \right)\to \exists !x\left(\phi \vee \psi \right)$$