Theorem 2ralbidv 2390

Description: Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)

Hypothesis

Ref	Expression
2ralbidv.1	|- ( ph -> ( ps <-> ch ) )

Assertion

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

Distinct variable groups:   ph, x   ph, y

Allowed substitution hints:   ps( x, y)   ch( x, y)   A( x, y)   B( x, y)

Proof of Theorem 2ralbidv

Step	Hyp	Ref	Expression
1		2ralbidv.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	ralbidv 2368	. 2 |- ( ph -> ( A. y e. B ps <-> A. y e. B ch ) )
3	2	ralbidv 2368	1 |- ( ph -> ( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353

This theorem is referenced by:  cbvral3v  2587  poeq1  4054  soeq1  4070  isoeq1  5461  isoeq2  5462  isoeq3  5463  smoeq  5928  elinp  6664  cauappcvgpr  6852  iseqcaopr2  9461  addcn2  10149  mulcn2  10151