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Theorem csbvarg 3280
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 2966 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vex 2961 . . . . . 6  |-  y  e. 
_V
3 df-csb 3254 . . . . . . 7  |-  [_ y  /  x ]_ x  =  { z  |  [. y  /  x ]. z  e.  x }
4 sbcel2gv 3223 . . . . . . . 8  |-  ( y  e.  _V  ->  ( [. y  /  x ]. z  e.  x  <->  z  e.  y ) )
54abbi1dv 2554 . . . . . . 7  |-  ( y  e.  _V  ->  { z  |  [. y  /  x ]. z  e.  x }  =  y )
63, 5syl5eq 2482 . . . . . 6  |-  ( y  e.  _V  ->  [_ y  /  x ]_ x  =  y )
72, 6ax-mp 5 . . . . 5  |-  [_ y  /  x ]_ x  =  y
87csbeq2i 3279 . . . 4  |-  [_ A  /  y ]_ [_ y  /  x ]_ x  = 
[_ A  /  y ]_ y
9 csbco 3262 . . . 4  |-  [_ A  /  y ]_ [_ y  /  x ]_ x  = 
[_ A  /  x ]_ x
10 df-csb 3254 . . . 4  |-  [_ A  /  y ]_ y  =  { z  |  [. A  /  y ]. z  e.  y }
118, 9, 103eqtr3i 2466 . . 3  |-  [_ A  /  x ]_ x  =  { z  |  [. A  /  y ]. z  e.  y }
12 sbcel2gv 3223 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  y ]. z  e.  y  <->  z  e.  A ) )
1312abbi1dv 2554 . . 3  |-  ( A  e.  _V  ->  { z  |  [. A  / 
y ]. z  e.  y }  =  A )
1411, 13syl5eq 2482 . 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 1653    e. wcel 1726   {cab 2424   _Vcvv 2958   [.wsbc 3163   [_csb 3253
This theorem is referenced by:  sbccsb2g  3282  csbfvg  5744  iuninc  24016  rusbcALT  27630  csbwrdg  28200  onfrALTlem5  28702  onfrALTlem4  28703  onfrALTlem5VD  29071  onfrALTlem4VD  29072  bnj110  29303  cdlemk40  31788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-csb 3254
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