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Theorem brab1 3920
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 2644 . . 3  |-  x  e. 
_V
2 breq1 3878 . . . 4  |-  ( z  =  y  ->  (
z R A  <->  y R A ) )
3 breq1 3878 . . . 4  |-  ( y  =  x  ->  (
y R A  <->  x R A ) )
42, 3sbcie2g 2894 . . 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 2863 . 2  |-  ( [. x  /  z ]. z R A  <->  x  e.  { z  |  z R A } )
75, 6bitr3i 185 1  |-  ( x R A  <->  x  e.  { z  |  z R A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1448   {cab 2086   _Vcvv 2641   [.wsbc 2862   class class class wbr 3875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sbc 2863  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876
This theorem is referenced by: (None)
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