Theorem ralbidv2

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997)

	Ref	Expression
Hypothesis	ralbidv2.1	|- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) )
Assertion	ralbidv2	|- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

Proof

Step	Hyp	Ref	Expression
1		ralbidv2.1	|- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) )
2	1	albidv	|- ( ph -> ( A. x ( x e. A -> ps ) <-> A. x ( x e. B -> ch ) ) )
3		df-ral	|- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
4		df-ral	|- ( A. x e. B ch <-> A. x ( x e. B -> ch ) )
5	2 3 4	3bitr4g	|- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )