Theorem 2reu2rex1

Description: Double restricted existential uniqueness implies double restricted existence. (Contributed by Thierry Arnoux, 4-Jul-2023)

	Ref	Expression
Assertion	2reu2rex1	$$⊢\exists !x\in A,y\in B\phi \to \exists x\in A\exists y\in B\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-2reu	$$⊢\exists !x\in A,y\in B\phi \leftrightarrow \left(\exists !x\in A\exists y\in B\phi \wedge \exists !y\in B\exists x\in A\phi \right)$$
2	1	simplbi	$$⊢\exists !x\in A,y\in B\phi \to \exists !x\in A\exists y\in B\phi$$
3		reurex	$$⊢\exists !x\in A\exists y\in B\phi \to \exists x\in A\exists y\in B\phi$$
4	2 3	syl	$$⊢\exists !x\in A,y\in B\phi \to \exists x\in A\exists y\in B\phi$$