Theorem cbveudavw

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

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

Proof

Step	Hyp	Ref	Expression
1		cbveudavw.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2	1	cbvexdvaw	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists y\chi \right)$$
3	1	cbvmodavw	$$⊢\phi \to \left({\exists }^{*}x\psi \leftrightarrow {\exists }^{*}y\chi \right)$$
4	2 3	anbi12d	$$⊢\phi \to \left(\left(\exists x\psi \wedge {\exists }^{*}x\psi \right)\leftrightarrow \left(\exists y\chi \wedge {\exists }^{*}y\chi \right)\right)$$
5		df-eu	$$⊢\exists !x\psi \leftrightarrow \left(\exists x\psi \wedge {\exists }^{*}x\psi \right)$$
6		df-eu	$$⊢\exists !y\chi \leftrightarrow \left(\exists y\chi \wedge {\exists }^{*}y\chi \right)$$
7	4 5 6	3bitr4g	$$⊢\phi \to \left(\exists !x\psi \leftrightarrow \exists !y\chi \right)$$