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Theorem rabeqbidv 2684
Description: Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
Hypotheses
Ref Expression
rabeqbidv.1  |-  ( ph  ->  A  =  B )
rabeqbidv.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
rabeqbidv  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hints:    ps( x)    ch( x)

Proof of Theorem rabeqbidv
StepHypRef Expression
1 rabeqbidv.1 . . 3  |-  ( ph  ->  A  =  B )
2 rabeq 2681 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  ps }  =  { x  e.  B  |  ps } )
31, 2syl 14 . 2  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ps } )
4 rabeqbidv.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
54rabbidv 2678 . 2  |-  ( ph  ->  { x  e.  B  |  ps }  =  {
x  e.  B  |  ch } )
63, 5eqtrd 2173 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1332   {crab 2421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426
This theorem is referenced by:  elfvmptrab1  5522  mpoxopoveq  6144  supeq123d  6885  phival  11923  dfphi2  11930  cldval  12305  neifval  12346  cnfval  12400  cnpfval  12401  cnprcl2k  12412  hmeofvalg  12509  ispsmet  12529  ismet  12550  isxmet  12551  blfvalps  12591  cncfval  12765
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