MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbvarg Unicode version

Theorem csbvarg 3121
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 2809 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vex 2804 . . . . . 6  |-  y  e. 
_V
3 df-csb 3095 . . . . . . 7  |-  [_ y  /  x ]_ x  =  { z  |  [. y  /  x ]. z  e.  x }
4 sbcel2gv 3064 . . . . . . . 8  |-  ( y  e.  _V  ->  ( [. y  /  x ]. z  e.  x  <->  z  e.  y ) )
54abbi1dv 2412 . . . . . . 7  |-  ( y  e.  _V  ->  { z  |  [. y  /  x ]. z  e.  x }  =  y )
63, 5syl5eq 2340 . . . . . 6  |-  ( y  e.  _V  ->  [_ y  /  x ]_ x  =  y )
72, 6ax-mp 8 . . . . 5  |-  [_ y  /  x ]_ x  =  y
87csbeq2i 3120 . . . 4  |-  [_ A  /  y ]_ [_ y  /  x ]_ x  = 
[_ A  /  y ]_ y
9 csbco 3103 . . . 4  |-  [_ A  /  y ]_ [_ y  /  x ]_ x  = 
[_ A  /  x ]_ x
10 df-csb 3095 . . . 4  |-  [_ A  /  y ]_ y  =  { z  |  [. A  /  y ]. z  e.  y }
118, 9, 103eqtr3i 2324 . . 3  |-  [_ A  /  x ]_ x  =  { z  |  [. A  /  y ]. z  e.  y }
12 sbcel2gv 3064 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  y ]. z  e.  y  <->  z  e.  A ) )
1312abbi1dv 2412 . . 3  |-  ( A  e.  _V  ->  { z  |  [. A  / 
y ]. z  e.  y }  =  A )
1411, 13syl5eq 2340 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ x  =  A )
151, 14syl 15 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   [.wsbc 3004   [_csb 3094
This theorem is referenced by:  sbccsb2g  3123  csbfvg  5554  iuninc  23174  rusbcALT  27742  onfrALTlem5  28606  onfrALTlem4  28607  onfrALTlem5VD  28977  onfrALTlem4VD  28978  bnj110  29206  cdlemk40  31728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005  df-csb 3095
  Copyright terms: Public domain W3C validator