Theorem ralbii2 2443

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 1446	. 2 |- ( A. x ( x e. A -> ph ) <-> A. x ( x e. B -> ps ) )
3		df-ral 2419	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4		df-ral 2419	. 2 |- ( A. x e. B ps <-> A. x ( x e. B -> ps ) )
5	2, 3, 4	3bitr4i 211	1 |- ( A. x e. A ph <-> A. x e. B ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   e. wcel 1480   A.wral 2414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425

This theorem depends on definitions:  df-bi 116  df-ral 2419

This theorem is referenced by:  raleqbii  2445  ralbiia  2447  ralrab  2840  raldifb  3211  raluz2  9367  ralrp  9456  isprm4  11789