Theorem ralbii2 2893

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 1620	. 2 |- ( A. x ( x e. A -> ph ) <-> A. x ( x e. B -> ps ) )
3		df-ral 2819	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4		df-ral 2819	. 2 |- ( A. x e. B ps <-> A. x ( x e. B -> ps ) )
5	2, 3, 4	3bitr4i 277	1 |- ( A. x e. A ph <-> A. x e. B ps )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1377   e. wcel 1767   A.wral 2814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612

This theorem depends on definitions:  df-bi 185  df-ral 2819

This theorem is referenced by:  ralbiia  2894  ralbii  2895  raleqbii  2909  ralrab  3265  raldifb  3644  raldifsni  4157  reusv2  4653  dfsup2  7902  iscard2  8357  acnnum  8433  dfac9  8516  dfacacn  8521  raluz2  11130  ralrp  11238  isprm4  14086  isdomn2  17747  isnrm2  19653  ismbl  21700  ellimc3  22046  dchrelbas2  23268  h1dei  26172  fnwe2lem2  30629  sdrgacs  30783