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Theorem rabbidva2 3096
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 2590 . 2  |-  ( ph  ->  { x  |  ( x  e.  A  /\  ps ) }  =  {
x  |  ( x  e.  B  /\  ch ) } )
3 df-rab 2813 . 2  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
4 df-rab 2813 . 2  |-  { x  e.  B  |  ch }  =  { x  |  ( x  e.  B  /\  ch ) }
52, 3, 43eqtr4g 2520 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   {crab 2808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-rab 2813
This theorem is referenced by:  extmptsuppeq  6916  dfac2a  8501  hashbclem  12485  isusgra0  24549  wwlkn0s  24907  wwlkextwrd  24930  rusgranumwlkl1  25149  rusgranumwlklem3  25153  numclwwlkovf2  25286  orvcgteel  28670  orvclteel  28675  mapdvalc  37753  mapdval4N  37756
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