Theorem 2ralbii 2478

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 2476	. 2 |- ( A. y e. B ph <-> A. y e. B ps )
3	2	ralbii 2476	1 |- ( A. x e. A A. y e. B ph <-> A. x e. A A. y e. B ps )

Syntax hints:   <-> wb 104   A.wral 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-ral 2453

This theorem is referenced by:  rmo4f  2928  ordsoexmid  4546  cnvsom  5154  fununi  5266  tpossym  6255  axpre-suploc  7864  issubm  12695  isbasis2g  12837