Theorem 2reucom

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

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

Proof

Step	Hyp	Ref	Expression
1		ancom	$$⊢\left(\exists !x\in A\exists y\in B\phi \wedge \exists !y\in B\exists x\in A\phi \right)\leftrightarrow \left(\exists !y\in B\exists x\in A\phi \wedge \exists !x\in A\exists y\in B\phi \right)$$
2		df-2reu	$$⊢\exists !x\in A,y\in B\phi \leftrightarrow \left(\exists !x\in A\exists y\in B\phi \wedge \exists !y\in B\exists x\in A\phi \right)$$
3		df-2reu	$$⊢\exists !y\in B,x\in A\phi \leftrightarrow \left(\exists !y\in B\exists x\in A\phi \wedge \exists !x\in A\exists y\in B\phi \right)$$
4	1 2 3	3bitr4i	$$⊢\exists !x\in A,y\in B\phi \leftrightarrow \exists !y\in B,x\in A\phi$$