Theorem cbvreudavw

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

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

Proof

Step	Hyp	Ref	Expression
1		cbvreudavw.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
3	2	adantl	$$⊢\left(\phi \wedge x=y\right)\to \left(x\in A\leftrightarrow y\in A\right)$$
4	3 1	anbi12d	$$⊢\left(\phi \wedge x=y\right)\to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(y\in A\wedge \chi \right)\right)$$
5	4	cbveudavw	$$⊢\phi \to \left(\exists !x\left(x\in A\wedge \psi \right)\leftrightarrow \exists !y\left(y\in A\wedge \chi \right)\right)$$
6		df-reu	$$⊢\exists !x\in A\psi \leftrightarrow \exists !x\left(x\in A\wedge \psi \right)$$
7		df-reu	$$⊢\exists !y\in A\chi \leftrightarrow \exists !y\left(y\in A\wedge \chi \right)$$
8	5 6 7	3bitr4g	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !y\in A\chi \right)$$