Theorem ralbii2 1671

Description: Inference adding different restricted universal quantifiers to each side of an equivalence.

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 999	. 2 |- (A.x(x e. A -> ph) <-> A.x(x e. B -> ps))
3		df-ral 1649	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
4		df-ral 1649	. 2 |- (A.x e. B ps <-> A.x(x e. B -> ps))
5	2, 3, 4	3bitr4 183	1 |- (A.x e. A ph <-> A.x e. B ps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   e. wcel 958  A.wral 1645

This theorem is referenced by:  ralbiia 1673  zmin 6219  ralrp 6289  raluz2 6443  clm4 7080  h1det 9473

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1649

Copyright terms: Public domain