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Theorem rabbidva2 3060
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 2587 . 2  |-  ( ph  ->  { x  |  ( x  e.  A  /\  ps ) }  =  {
x  |  ( x  e.  B  /\  ch ) } )
3 df-rab 2804 . 2  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
4 df-rab 2804 . 2  |-  { x  e.  B  |  ch }  =  { x  |  ( x  e.  B  /\  ch ) }
52, 3, 43eqtr4g 2517 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2436   {crab 2799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-rab 2804
This theorem is referenced by:  orvcgteel  26986  orvclteel  26991  wwlkn0s  30479
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