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

Theorem csbvarg 3223
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 2909 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vex 2904 . . . . . 6  |-  y  e. 
_V
3 df-csb 3197 . . . . . . 7  |-  [_ y  /  x ]_ x  =  { z  |  [. y  /  x ]. z  e.  x }
4 sbcel2gv 3166 . . . . . . . 8  |-  ( y  e.  _V  ->  ( [. y  /  x ]. z  e.  x  <->  z  e.  y ) )
54abbi1dv 2505 . . . . . . 7  |-  ( y  e.  _V  ->  { z  |  [. y  /  x ]. z  e.  x }  =  y )
63, 5syl5eq 2433 . . . . . 6  |-  ( y  e.  _V  ->  [_ y  /  x ]_ x  =  y )
72, 6ax-mp 8 . . . . 5  |-  [_ y  /  x ]_ x  =  y
87csbeq2i 3222 . . . 4  |-  [_ A  /  y ]_ [_ y  /  x ]_ x  = 
[_ A  /  y ]_ y
9 csbco 3205 . . . 4  |-  [_ A  /  y ]_ [_ y  /  x ]_ x  = 
[_ A  /  x ]_ x
10 df-csb 3197 . . . 4  |-  [_ A  /  y ]_ y  =  { z  |  [. A  /  y ]. z  e.  y }
118, 9, 103eqtr3i 2417 . . 3  |-  [_ A  /  x ]_ x  =  { z  |  [. A  /  y ]. z  e.  y }
12 sbcel2gv 3166 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  y ]. z  e.  y  <->  z  e.  A ) )
1312abbi1dv 2505 . . 3  |-  ( A  e.  _V  ->  { z  |  [. A  / 
y ]. z  e.  y }  =  A )
1411, 13syl5eq 2433 . 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 1649    e. wcel 1717   {cab 2375   _Vcvv 2901   [.wsbc 3106   [_csb 3196
This theorem is referenced by:  sbccsb2g  3225  csbfvg  5683  iuninc  23857  rusbcALT  27310  onfrALTlem5  27973  onfrALTlem4  27974  onfrALTlem5VD  28340  onfrALTlem4VD  28341  bnj110  28569  cdlemk40  31033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-sbc 3107  df-csb 3197
  Copyright terms: Public domain W3C validator