Theorem cbvriotadavw

Description: Change bound variable in a restricted description binder. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvriotadavw.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
Assertion	cbvriotadavw	|- ( ph -> ( iota_ x e. A ps ) = ( iota_ y e. A ch ) )

Proof

Step	Hyp	Ref	Expression
1		cbvriotadavw.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2		eleq1w	|- ( x = y -> ( x e. A <-> y e. A ) )
3	2	adantl	|- ( ( ph /\ x = y ) -> ( x e. A <-> y e. A ) )
4	3 1	anbi12d	|- ( ( ph /\ x = y ) -> ( ( x e. A /\ ps ) <-> ( y e. A /\ ch ) ) )
5	4	cbviotadavw	|- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota y ( y e. A /\ ch ) ) )
6		df-riota	|- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
7		df-riota	|- ( iota_ y e. A ch ) = ( iota y ( y e. A /\ ch ) )
8	5 6 7	3eqtr4g	|- ( ph -> ( iota_ x e. A ps ) = ( iota_ y e. A ch ) )