Theorem 2reu2rex

Description: Double restricted existential uniqueness, analogous to 2eu2ex . (Contributed by Alexander van der Vekens, 25-Jun-2017)

	Ref	Expression
Assertion	2reu2rex	$$⊢\exists !x\in A\exists !y\in B\phi \to \exists x\in A\exists y\in B\phi$$

Proof

Step	Hyp	Ref	Expression
1		reurex	$$⊢\exists !x\in A\exists !y\in B\phi \to \exists x\in A\exists !y\in B\phi$$
2		reurex	$$⊢\exists !y\in B\phi \to \exists y\in B\phi$$
3	2	reximi	$$⊢\exists x\in A\exists !y\in B\phi \to \exists x\in A\exists y\in B\phi$$
4	1 3	syl	$$⊢\exists !x\in A\exists !y\in B\phi \to \exists x\in A\exists y\in B\phi$$