Theorem 2ralbii 2541

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 2539	. 2 |- ( A. y e. B ph <-> A. y e. B ps )
3	2	ralbii 2539	1 |- ( A. x e. A A. y e. B ph <-> A. x e. A A. y e. B ps )

Syntax hints:   <-> wb 105   A.wral 2511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-4 1559  ax-17 1575

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-ral 2516

This theorem is referenced by:  rmo4f  3005  ordsoexmid  4666  cnvsom  5287  fununi  5405  tpossym  6485  axpre-suploc  8182  issubm  13635  isbasis2g  14856  ivthdich  15464