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Theorem csbvarg 2877
Description: The proper substitution of a class for setvar 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 2566 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vex 2560 . . . . . 6  |-  y  e. 
_V
3 df-csb 2853 . . . . . . 7  |-  [_ y  /  x ]_ x  =  { z  |  [. y  /  x ]. z  e.  x }
4 sbcel2gv 2822 . . . . . . . 8  |-  ( y  e.  _V  ->  ( [. y  /  x ]. z  e.  x  <->  z  e.  y ) )
54abbi1dv 2157 . . . . . . 7  |-  ( y  e.  _V  ->  { z  |  [. y  /  x ]. z  e.  x }  =  y )
63, 5syl5eq 2084 . . . . . 6  |-  ( y  e.  _V  ->  [_ y  /  x ]_ x  =  y )
72, 6ax-mp 7 . . . . 5  |-  [_ y  /  x ]_ x  =  y
87csbeq2i 2876 . . . 4  |-  [_ A  /  y ]_ [_ y  /  x ]_ x  = 
[_ A  /  y ]_ y
9 csbco 2861 . . . 4  |-  [_ A  /  y ]_ [_ y  /  x ]_ x  = 
[_ A  /  x ]_ x
10 df-csb 2853 . . . 4  |-  [_ A  /  y ]_ y  =  { z  |  [. A  /  y ]. z  e.  y }
118, 9, 103eqtr3i 2068 . . 3  |-  [_ A  /  x ]_ x  =  { z  |  [. A  /  y ]. z  e.  y }
12 sbcel2gv 2822 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  y ]. z  e.  y  <->  z  e.  A ) )
1312abbi1dv 2157 . . 3  |-  ( A  e.  _V  ->  { z  |  [. A  / 
y ]. z  e.  y }  =  A )
1411, 13syl5eq 2084 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ x  =  A )
151, 14syl 14 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393   {cab 2026   _Vcvv 2557   [.wsbc 2764   [_csb 2852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853
This theorem is referenced by:  sbccsb2g  2879  csbfvg  5211  bj-sels  10007
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