Theorem cbvrmow

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 16-Jun-2017) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvrmow.1	$$⊢Ⅎy\phi$$
	cbvrmow.2	$$⊢Ⅎx\psi$$
	cbvrmow.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvrmow	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvrmow.1	$$⊢Ⅎy\phi$$
2		cbvrmow.2	$$⊢Ⅎx\psi$$
3		cbvrmow.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	cbvreuw	$$⊢\exists !x\in A\phi \leftrightarrow \exists !y\in A\psi$$
6	4 5	imbi12i	$$⊢\left(\exists x\in A\phi \to \exists !x\in A\phi \right)\leftrightarrow \left(\exists y\in A\psi \to \exists !y\in A\psi \right)$$
7		rmo5	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow \left(\exists x\in A\phi \to \exists !x\in A\phi \right)$$
8		rmo5	$$⊢{\exists }^{*}y\in A\psi \leftrightarrow \left(\exists y\in A\psi \to \exists !y\in A\psi \right)$$
9	6 7 8	3bitr4i	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}y\in A\psi$$