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Theorem csbvarg 3270
Description: The proper substitution of a class for set variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbvarg  |-  ( A  e.  V  ->  [_ A  /  x ]_ x  =  A )

Proof of Theorem csbvarg
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2956 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vex 2951 . . . . . 6  |-  y  e. 
_V
3 df-csb 3244 . . . . . . 7  |-  [_ y  /  x ]_ x  =  { z  |  [. y  /  x ]. z  e.  x }
4 sbcel2gv 3213 . . . . . . . 8  |-  ( y  e.  _V  ->  ( [. y  /  x ]. z  e.  x  <->  z  e.  y ) )
54abbi1dv 2551 . . . . . . 7  |-  ( y  e.  _V  ->  { z  |  [. y  /  x ]. z  e.  x }  =  y )
63, 5syl5eq 2479 . . . . . 6  |-  ( y  e.  _V  ->  [_ y  /  x ]_ x  =  y )
72, 6ax-mp 8 . . . . 5  |-  [_ y  /  x ]_ x  =  y
87csbeq2i 3269 . . . 4  |-  [_ A  /  y ]_ [_ y  /  x ]_ x  = 
[_ A  /  y ]_ y
9 csbco 3252 . . . 4  |-  [_ A  /  y ]_ [_ y  /  x ]_ x  = 
[_ A  /  x ]_ x
10 df-csb 3244 . . . 4  |-  [_ A  /  y ]_ y  =  { z  |  [. A  /  y ]. z  e.  y }
118, 9, 103eqtr3i 2463 . . 3  |-  [_ A  /  x ]_ x  =  { z  |  [. A  /  y ]. z  e.  y }
12 sbcel2gv 3213 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  y ]. z  e.  y  <->  z  e.  A ) )
1312abbi1dv 2551 . . 3  |-  ( A  e.  _V  ->  { z  |  [. A  / 
y ]. z  e.  y }  =  A )
1411, 13syl5eq 2479 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ x  =  A )
151, 14syl 16 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948   [.wsbc 3153   [_csb 3243
This theorem is referenced by:  sbccsb2g  3272  csbfvg  5733  iuninc  24003  rusbcALT  27607  onfrALTlem5  28565  onfrALTlem4  28566  onfrALTlem5VD  28934  onfrALTlem4VD  28935  bnj110  29166  cdlemk40  31651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154  df-csb 3244
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