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Theorem rabeqbidv 2597
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 2596 . . 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 2594 . 2  |-  ( ph  ->  { x  e.  B  |  ps }  =  {
x  e.  B  |  ch } )
63, 5eqtrd 2114 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   {crab 2353
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358
This theorem is referenced by:  mpt2xopoveq  5883  supeq123d  6453
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