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Theorem csbex 3092
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 3091 . . 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 1535 1  |-  [_ A  /  x ]_ B  e. 
_V
Colors of variables: wff set class
Syntax hints:   A.wal 1527    e. wcel 1684   _Vcvv 2788   [_csb 3081
This theorem is referenced by:  dfmpt2  6209  cantnfdm  7365  cantnff  7375  ruclem1  12509  pcmpt  12940  cidffn  13580  natffn  13823  fnxpc  13950  evlfcl  13996  odf  14852  itgfsum  19181  vmaf  20357  bpolylem  24783  aomclem6  27156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
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