Theorem ralbii2 2643

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 1566	. 2 |- (A.x(x e. A -> ph) <-> A.x(x e. B -> ps))
3		df-ral 2620	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
4		df-ral 2620	. 2 |- (A.x e. B ps <-> A.x(x e. B -> ps))
5	2, 3, 4	3bitr4i 268	1 |- (A.x e. A ph <-> A.x e. B ps)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   e. wcel 1710  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ral 2620

This theorem is referenced by:  raleqbii  2645  ralbiia  2647  ralrab  2999