Theorem ralbii2

Description: Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005)

	Ref	Expression
Hypothesis	ralbii2.1	|- ( ( x e. A -> ph ) <-> ( x e. B -> ps ) )
Assertion	ralbii2	|- ( A. x e. A ph <-> A. x e. B ps )

Proof

Step	Hyp	Ref	Expression
1		ralbii2.1	|- ( ( x e. A -> ph ) <-> ( x e. B -> ps ) )
2	1	albii	|- ( A. x ( x e. A -> ph ) <-> A. x ( x e. B -> ps ) )
3		df-ral	|- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4		df-ral	|- ( A. x e. B ps <-> A. x ( x e. B -> ps ) )
5	2 3 4	3bitr4i	|- ( A. x e. A ph <-> A. x e. B ps )