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Theorem brab1 4068
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 2791 . . 3  |-  x  e. 
_V
2 breq1 4026 . . . 4  |-  ( z  =  y  ->  (
z R A  <->  y R A ) )
3 breq1 4026 . . . 4  |-  ( y  =  x  ->  (
y R A  <->  x R A ) )
42, 3sbcie2g 3024 . . 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 2992 . 2  |-  ( [. x  /  z ]. z R A  <->  x  e.  { z  |  z R A } )
75, 6bitr3i 242 1  |-  ( x R A  <->  x  e.  { z  |  z R A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684   {cab 2269   _Vcvv 2788   [.wsbc 2991   class class class wbr 4023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024
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