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Theorem rabeqbidva 2797
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 2792 . 2  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  A  |  ch } )
3 rabeqbidva.1 . . 3  |-  ( ph  ->  A  =  B )
4 rabeq 2795 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  ch }  =  { x  e.  B  |  ch } )
53, 4syl 15 . 2  |-  ( ph  ->  { x  e.  A  |  ch }  =  {
x  e.  B  |  ch } )
62, 5eqtrd 2328 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560
This theorem is referenced by:  natpropd  13866  gsumpropd2lem  23394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565
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