Theorem 2ralbii 2332

Description: Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)

Hypothesis

Ref	Expression
ralbii.1	|- ( ph <-> ps )

Assertion

Ref	Expression
2ralbii	|- ( A. x e. A A. y e. B ph <-> A. x e. A A. y e. B ps )

Proof of Theorem 2ralbii

Step	Hyp	Ref	Expression
1		ralbii.1	. . 3 |- ( ph <-> ps )
2	1	ralbii 2330	. 2 |- ( A. y e. B ph <-> A. y e. B ps )
3	2	ralbii 2330	1 |- ( A. x e. A A. y e. B ph <-> A. x e. A A. y e. B ps )

Syntax hints:   <-> wb 98   A.wral 2306

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 1336  ax-gen 1338  ax-4 1400  ax-17 1419

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-ral 2311

This theorem is referenced by:  ordsoexmid  4286  cnvsom  4861  fununi  4967  tpossym  5891