Theorem cbvreuw

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 15-Oct-2016) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024) Avoid ax-10 . (Revised by Wolf Lammen, 10-Dec-2024)

	Ref	Expression
Hypotheses	cbvreuw.1	$$⊢Ⅎy\phi$$
	cbvreuw.2	$$⊢Ⅎx\psi$$
	cbvreuw.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvreuw	$$⊢\exists !x\in A\phi \leftrightarrow \exists !y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvreuw.1	$$⊢Ⅎy\phi$$
2		cbvreuw.2	$$⊢Ⅎx\psi$$
3		cbvreuw.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	1 2 3	cbvrexw	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\psi$$
5	1 2 3	cbvrmow	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}y\in A\psi$$
6	4 5	anbi12i	$$⊢\left(\exists x\in A\phi \wedge {\exists }^{*}x\in A\phi \right)\leftrightarrow \left(\exists y\in A\psi \wedge {\exists }^{*}y\in A\psi \right)$$
7		reu5	$$⊢\exists !x\in A\phi \leftrightarrow \left(\exists x\in A\phi \wedge {\exists }^{*}x\in A\phi \right)$$
8		reu5	$$⊢\exists !y\in A\psi \leftrightarrow \left(\exists y\in A\psi \wedge {\exists }^{*}y\in A\psi \right)$$
9	6 7 8	3bitr4i	$$⊢\exists !x\in A\phi \leftrightarrow \exists !y\in A\psi$$