Theorem ralbii2 2328

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 1356	. 2 |- (A.x(x e. A -> ph) <-> A.x(x e. B -> ps))
3		df-ral 2305	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
4		df-ral 2305	. 2 |- (A.x e. B ps <-> A.x(x e. B -> ps))
5	2, 3, 4	3bitr4i 201	1 |- (A.x e. A ph <-> A.x e. B ps)

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240   e. wcel 1390  A.wral 2300

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335

This theorem depends on definitions:  df-bi 110  df-ral 2305

This theorem is referenced by:  raleqbii  2330  ralbiia  2332  ralrab  2696  raldifb  3077  raluz2  8298  ralrp  8379