Theorem 2reu2reu2

Description: Double restricted existential uniqueness implies two nested restricted existential uniqueness. (Contributed by AV, 5-Jul-2023)

	Ref	Expression
Assertion	2reu2reu2	$$⊢\exists !x\in A,y\in B\phi \to \exists !x\in A\exists !y\in B\phi$$

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		2rexreu	$$⊢\left(\exists !x\in A\exists y\in B\phi \wedge \exists !y\in B\exists x\in A\phi \right)\to \exists !x\in A\exists !y\in B\phi$$
3	1 2	sylbi	$$⊢\exists !x\in A,y\in B\phi \to \exists !x\in A\exists !y\in B\phi$$