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Theorem brab1 3838
Description: Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
Assertion
Ref Expression
brab1  |-  ( x R A  <->  x  e.  { z  |  z R A } )
Distinct variable groups:    z, A    z, R
Allowed substitution hints:    A( x)    R( x)

Proof of Theorem brab1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2605 . . 3  |-  x  e. 
_V
2 breq1 3796 . . . 4  |-  ( z  =  y  ->  (
z R A  <->  y R A ) )
3 breq1 3796 . . . 4  |-  ( y  =  x  ->  (
y R A  <->  x R A ) )
42, 3sbcie2g 2848 . . 3  |-  ( x  e.  _V  ->  ( [. x  /  z ]. z R A  <->  x R A ) )
51, 4ax-mp 7 . 2  |-  ( [. x  /  z ]. z R A  <->  x R A )
6 df-sbc 2817 . 2  |-  ( [. x  /  z ]. z R A  <->  x  e.  { z  |  z R A } )
75, 6bitr3i 184 1  |-  ( x R A  <->  x  e.  { z  |  z R A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1434   {cab 2068   _Vcvv 2602   [.wsbc 2816   class class class wbr 3793
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  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-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794
This theorem is referenced by: (None)
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