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Theorem rabeqbidva 2606
Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
rabeqbidva.1  |-  ( ph  ->  A  =  B )
rabeqbidva.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rabeqbidva  |-  ( 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 rabeqbidva
StepHypRef Expression
1 rabeqbidva.2 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
21rabbidva 2598 . 2  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  A  |  ch } )
3 rabeqbidva.1 . . 3  |-  ( ph  ->  A  =  B )
4 rabeq 2602 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  ch }  =  { x  e.  B  |  ch } )
53, 4syl 14 . 2  |-  ( ph  ->  { x  e.  A  |  ch }  =  {
x  e.  B  |  ch } )
62, 5eqtrd 2115 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {crab 2357
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rab 2362
This theorem is referenced by: (None)
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