Theorem 2ralbii 2558

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 2556	. 2 |- ( A. y e. B ph <-> A. y e. B ps )
3	2	ralbii 2556	1 |- ( A. x e. A A. y e. B ph <-> A. x e. A A. y e. B ps )

This proof depends on syntax axioms:   <-> wb 105   A.wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-ral 2533

This theorem is used by:  rmo4f  3024  ordsoexmid  4709  cnvsom  5331  fununi  5449  tpossym  6547  axpre-suploc  8269  issubm  13779  isbasis2g  15146  ivthdich  15754