Theorem cbvriotavw2

Description: Change bound variable and domain in a restricted description binder, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvriotavw2.1	$$⊢x=y\to A=B$$
	cbvriotavw2.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvriotavw2	$$⊢\left(\iota x\in A|\phi \right)=\left(\iota y\in B|\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvriotavw2.1	$$⊢x=y\to A=B$$
2		cbvriotavw2.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3		id	$$⊢x=y\to x=y$$
4	3 1	eleq12d	$$⊢x=y\to \left(x\in A\leftrightarrow y\in B\right)$$
5	4 2	anbi12d	$$⊢x=y\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(y\in B\wedge \psi \right)\right)$$
6	5	cbviotavw	$$⊢\left(\iota x|\left(x\in A\wedge \phi \right)\right)=\left(\iota y|\left(y\in B\wedge \psi \right)\right)$$
7		df-riota	$$⊢\left(\iota x\in A|\phi \right)=\left(\iota x|\left(x\in A\wedge \phi \right)\right)$$
8		df-riota	$$⊢\left(\iota y\in B|\psi \right)=\left(\iota y|\left(y\in B\wedge \psi \right)\right)$$
9	6 7 8	3eqtr4i	$$⊢\left(\iota x\in A|\phi \right)=\left(\iota y\in B|\psi \right)$$