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Theorem brab1 4199
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 2903 . . 3  |-  x  e. 
_V
2 breq1 4157 . . . 4  |-  ( z  =  y  ->  (
z R A  <->  y R A ) )
3 breq1 4157 . . . 4  |-  ( y  =  x  ->  (
y R A  <->  x R A ) )
42, 3sbcie2g 3138 . . 3  |-  ( x  e.  _V  ->  ( [. x  /  z ]. z R A  <->  x R A ) )
51, 4ax-mp 8 . 2  |-  ( [. x  /  z ]. z R A  <->  x R A )
6 df-sbc 3106 . 2  |-  ( [. x  /  z ]. z R A  <->  x  e.  { z  |  z R A } )
75, 6bitr3i 243 1  |-  ( x R A  <->  x  e.  { z  |  z R A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1717   {cab 2374   _Vcvv 2900   [.wsbc 3105   class class class wbr 4154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155
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