Theorem ralbii2 2540

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 1516	. 2 |- ( A. x ( x e. A -> ph ) <-> A. x ( x e. B -> ps ) )
3		df-ral 2513	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4		df-ral 2513	. 2 |- ( A. x e. B ps <-> A. x ( x e. B -> ps ) )
5	2, 3, 4	3bitr4i 212	1 |- ( A. x e. A ph <-> A. x e. B ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   e. wcel 2200   A.wral 2508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495

This theorem depends on definitions:  df-bi 117  df-ral 2513

This theorem is referenced by:  raleqbii  2542  ralbiia  2544  ralrab  2964  raldifb  3344  raluz2  9770  ralrp  9867  isprm4  12636