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Theorem brab1 4052
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 2742 . . 3  |-  x  e. 
_V
2 breq1 4008 . . . 4  |-  ( z  =  y  ->  (
z R A  <->  y R A ) )
3 breq1 4008 . . . 4  |-  ( y  =  x  ->  (
y R A  <->  x R A ) )
42, 3sbcie2g 2998 . . 3  |-  ( x  e.  _V  ->  ( [. x  /  z ]. z R A  <->  x R A ) )
51, 4ax-mp 5 . 2  |-  ( [. x  /  z ]. z R A  <->  x R A )
6 df-sbc 2965 . 2  |-  ( [. x  /  z ]. z R A  <->  x  e.  { z  |  z R A } )
75, 6bitr3i 186 1  |-  ( x R A  <->  x  e.  { z  |  z R A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    e. wcel 2148   {cab 2163   _Vcvv 2739   [.wsbc 2964   class class class wbr 4005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006
This theorem is referenced by: (None)
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