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Theorem rabbidva 2593
Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
Hypothesis
Ref Expression
rabbidva.1  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rabbidva  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  A  |  ch } )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)

Proof of Theorem rabbidva
StepHypRef Expression
1 rabbidva.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
21ralrimiva 2435 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  <->  ch ) )
3 rabbi 2532 . 2  |-  ( A. x  e.  A  ( ps 
<->  ch )  <->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch } )
42, 3sylib 120 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  A  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   {crab 2353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-ral 2354  df-rab 2358
This theorem is referenced by:  rabbidv  2594  rabeqbidva  2598  rabbi2dva  3175  rabxfrd  4221  onsucmin  4253  seinxp  4431  fniniseg2  5315  fnniniseg2  5316  f1oresrab  5355  dfinfre  8090  minmax  10239  gcdass  10537  lcmass  10600
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