Theorem ralbii2 2376

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

Hypothesis

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

Assertion

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

Proof of Theorem ralbii2

Step	Hyp	Ref	Expression
1		ralbii2.1	. . 3 |- ( ( x e. A -> ph ) <-> ( x e. B -> ps ) )
2	1	albii 1399	. 2 |- ( A. x ( x e. A -> ph ) <-> A. x ( x e. B -> ps ) )
3		df-ral 2353	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4		df-ral 2353	. 2 |- ( A. x e. B ps <-> A. x ( x e. B -> ps ) )
5	2, 3, 4	3bitr4i 210	1 |- ( A. x e. A ph <-> A. x e. B ps )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378

This theorem depends on definitions:  df-bi 115  df-ral 2353

This theorem is referenced by:  raleqbii  2378  ralbiia  2380  ralrab  2753  raldifb  3112  raluz2  8667  ralrp  8755  isprm4  10501