Theorem raleqbidv 2574

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)

Hypotheses

Ref	Expression
raleqbidv.1	|- ( ph -> A = B )
raleqbidv.2	|- ( ph -> ( ps <-> ch ) )

Assertion

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

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hints:   ps( x)   ch( x)

Proof of Theorem raleqbidv

Step	Hyp	Ref	Expression
1		raleqbidv.1	. . 3 |- ( ph -> A = B )
2	1	raleqdv 2568	. 2 |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) )
3		raleqbidv.2	. . 3 |- ( ph -> ( ps <-> ch ) )
4	3	ralbidv 2380	. 2 |- ( ph -> ( A. x e. B ps <-> A. x e. B ch ) )
5	2, 4	bitrd 186	1 |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364

This theorem is referenced by:  ofrfval  5864  fmpt2x  5970  tfrlemi1  6097  supeq123d  6686  cvg1nlemcau  10417  cvg1nlemres  10418  cau3lem  10547  fsum2dlemstep  10828  fisumcom2  10832  istopg  11596  sscoll2  11883