Theorem 2ralbidv 2402

Description: Formula-building rule for restricted universal quantifiers (deduction form). (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 2380	. 2 |- ( ph -> ( A. y e. B ps <-> A. y e. B ch ) )
3	2	ralbidv 2380	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 2359

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 1381  ax-gen 1383  ax-4 1445  ax-17 1464

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-ral 2364

This theorem is referenced by:  cbvral3v  2600  poeq1  4126  soeq1  4142  isoeq1  5580  isoeq2  5581  isoeq3  5582  smoeq  6055  xpf1o  6560  elinp  7033  cauappcvgpr  7221  iseqcaopr2  9911  addcn2  10699  mulcn2  10701  elcncf  11629