Theorem riotaeqbidva

Description: Equivalent wff's yield equal restricted definition binders (deduction form). ( raleqbidva analog.) (Contributed by Thierry Arnoux, 29-Jan-2025)

	Ref	Expression
Hypotheses	riotaeqbidva.1	|- ( ph -> A = B )
	riotaeqbidva.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
Assertion	riotaeqbidva	|- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ch ) )

Proof

Step	Hyp	Ref	Expression
1		riotaeqbidva.1	|- ( ph -> A = B )
2		riotaeqbidva.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	riotabidva	|- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )
4	1	riotaeqdv	|- ( ph -> ( iota_ x e. A ch ) = ( iota_ x e. B ch ) )
5	3 4	eqtrd	|- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ch ) )