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Theorem rabeqbidva 3057
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 3052 . 2  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  A  |  ch } )
3 rabeqbidva.1 . . 3  |-  ( ph  ->  A  =  B )
4 rabeq 3055 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  ch }  =  { x  e.  B  |  ch } )
53, 4syl 17 . 2  |-  ( ph  ->  { x  e.  A  |  ch }  =  {
x  e.  B  |  ch } )
62, 5eqtrd 2445 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   {crab 2760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-rab 2765
This theorem is referenced by:  natpropd  15591  gsumpropd2lem  16226  eengv  24711  elntg  24716  domnmsuppn0  38486
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