Theorem riotaeqbidv

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011)

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

Proof

Step	Hyp	Ref	Expression
1		riotaeqbidv.1	|- ( ph -> A = B )
2		riotaeqbidv.2	|- ( ph -> ( ps <-> ch ) )
3	2	riotabidv	|- ( 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 ) )