Theorem cbvreuvw

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 5-Apr-2004) (Revised by GG, 30-Sep-2024)

	Ref	Expression
Hypothesis	cbvrmovw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvreuvw	$$⊢\exists !x\in A\phi \leftrightarrow \exists !y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvrmovw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
3	2 1	anbi12d	$$⊢x=y\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(y\in A\wedge \psi \right)\right)$$
4	3	cbveuvw	$$⊢\exists !x\left(x\in A\wedge \phi \right)\leftrightarrow \exists !y\left(y\in A\wedge \psi \right)$$
5		df-reu	$$⊢\exists !x\in A\phi \leftrightarrow \exists !x\left(x\in A\wedge \phi \right)$$
6		df-reu	$$⊢\exists !y\in A\psi \leftrightarrow \exists !y\left(y\in A\wedge \psi \right)$$
7	4 5 6	3bitr4i	$$⊢\exists !x\in A\phi \leftrightarrow \exists !y\in A\psi$$