Theorem cbvreudavw2

Description: Change bound variable and quantifier domain in the restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvreudavw2.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
	cbvreudavw2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
Assertion	cbvreudavw2	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !y\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvreudavw2.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		cbvreudavw2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
3		simpr	$$⊢\left(\phi \wedge x=y\right)\to x=y$$
4	3 2	eleq12d	$$⊢\left(\phi \wedge x=y\right)\to \left(x\in A\leftrightarrow y\in B\right)$$
5	4 1	anbi12d	$$⊢\left(\phi \wedge x=y\right)\to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(y\in B\wedge \chi \right)\right)$$
6	5	cbveudavw	$$⊢\phi \to \left(\exists !x\left(x\in A\wedge \psi \right)\leftrightarrow \exists !y\left(y\in B\wedge \chi \right)\right)$$
7		df-reu	$$⊢\exists !x\in A\psi \leftrightarrow \exists !x\left(x\in A\wedge \psi \right)$$
8		df-reu	$$⊢\exists !y\in B\chi \leftrightarrow \exists !y\left(y\in B\wedge \chi \right)$$
9	6 7 8	3bitr4g	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !y\in B\chi \right)$$