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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
StepHypRef Expression
1 raleqbidv.1 . . 3  |-  ( ph  ->  A  =  B )
21raleqdv 2568 . 2  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ps )
)
3 raleqbidv.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
43ralbidv 2380 . 2  |-  ( ph  ->  ( A. x  e.  B  ps  <->  A. x  e.  B  ch )
)
52, 4bitrd 186 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
)
Colors of variables: wff set class
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
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