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Theorem csbex 3105
Description: The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypotheses
Ref Expression
csbex.1  |-  A  e. 
_V
csbex.2  |-  B  e. 
_V
Assertion
Ref Expression
csbex  |-  [_ A  /  x ]_ B  e. 
_V

Proof of Theorem csbex
StepHypRef Expression
1 csbex.1 . . 3  |-  A  e. 
_V
2 csbexg 3104 . . 3  |-  ( ( A  e.  _V  /\  A. x  B  e.  _V )  ->  [_ A  /  x ]_ B  e.  _V )
31, 2mpan 651 . 2  |-  ( A. x  B  e.  _V  ->  [_ A  /  x ]_ B  e.  _V )
4 csbex.2 . 2  |-  B  e. 
_V
53, 4mpg 1538 1  |-  [_ A  /  x ]_ B  e. 
_V
Colors of variables: wff set class
Syntax hints:   A.wal 1530    e. wcel 1696   _Vcvv 2801   [_csb 3094
This theorem is referenced by:  dfmpt2  6225  cantnfdm  7381  cantnff  7391  ruclem1  12525  pcmpt  12956  cidffn  13596  natffn  13839  fnxpc  13966  evlfcl  14012  odf  14868  itgfsum  19197  vmaf  20373  bpolylem  24855  aomclem6  27259
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
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