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Theorem rabbidva2 2645
Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
Hypothesis
Ref Expression
rabbidva2.1  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ch ) ) )
Assertion
Ref Expression
rabbidva2  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)    B( x)

Proof of Theorem rabbidva2
StepHypRef Expression
1 rabbidva2.1 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ch ) ) )
21abbidv 2233 . 2  |-  ( ph  ->  { x  |  ( x  e.  A  /\  ps ) }  =  {
x  |  ( x  e.  B  /\  ch ) } )
3 df-rab 2400 . 2  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
4 df-rab 2400 . 2  |-  { x  e.  B  |  ch }  =  { x  |  ( x  e.  B  /\  ch ) }
52, 3, 43eqtr4g 2173 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   {cab 2101   {crab 2395
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-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-rab 2400
This theorem is referenced by: (None)
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