Theorem 2rexreu

Description: Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu . (Contributed by Alexander van der Vekens, 25-Jun-2017)

	Ref	Expression
Assertion	2rexreu	$$⊢\left(\exists !x\in A\exists y\in B\phi \wedge \exists !y\in B\exists x\in A\phi \right)\to \exists !x\in A\exists !y\in B\phi$$

Proof

Step	Hyp	Ref	Expression
1		reurmo	$$⊢\exists !x\in A\exists y\in B\phi \to {\exists }^{*}x\in A\exists y\in B\phi$$
2		reurex	$$⊢\exists !y\in B\phi \to \exists y\in B\phi$$
3	2	rmoimi	$$⊢{\exists }^{*}x\in A\exists y\in B\phi \to {\exists }^{*}x\in A\exists !y\in B\phi$$
4	1 3	syl	$$⊢\exists !x\in A\exists y\in B\phi \to {\exists }^{*}x\in A\exists !y\in B\phi$$
5		2reurex	$$⊢\exists !y\in B\exists x\in A\phi \to \exists x\in A\exists !y\in B\phi$$
6	4 5	anim12ci	$$⊢\left(\exists !x\in A\exists y\in B\phi \wedge \exists !y\in B\exists x\in A\phi \right)\to \left(\exists x\in A\exists !y\in B\phi \wedge {\exists }^{*}x\in A\exists !y\in B\phi \right)$$
7		reu5	$$⊢\exists !x\in A\exists !y\in B\phi \leftrightarrow \left(\exists x\in A\exists !y\in B\phi \wedge {\exists }^{*}x\in A\exists !y\in B\phi \right)$$
8	6 7	sylibr	$$⊢\left(\exists !x\in A\exists y\in B\phi \wedge \exists !y\in B\exists x\in A\phi \right)\to \exists !x\in A\exists !y\in B\phi$$