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 -> A = B )
	cbvriotavw2.2	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvriotavw2	|- ( iota_ x e. A ph ) = ( iota_ y e. B ps )

Proof

Step	Hyp	Ref	Expression
1		cbvriotavw2.1	|- ( x = y -> A = B )
2		cbvriotavw2.2	|- ( x = y -> ( ph <-> ps ) )
3		id	|- ( x = y -> x = y )
4	3 1	eleq12d	|- ( x = y -> ( x e. A <-> y e. B ) )
5	4 2	anbi12d	|- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. B /\ ps ) ) )
6	5	cbviotavw	|- ( iota x ( x e. A /\ ph ) ) = ( iota y ( y e. B /\ ps ) )
7		df-riota	|- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
8		df-riota	|- ( iota_ y e. B ps ) = ( iota y ( y e. B /\ ps ) )
9	6 7 8	3eqtr4i	|- ( iota_ x e. A ph ) = ( iota_ y e. B ps )