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Theorem csbexa 3968
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
csbexa.1  |-  A  e. 
_V
csbexa.2  |-  B  e. 
_V
Assertion
Ref Expression
csbexa  |-  [_ A  /  x ]_ B  e. 
_V

Proof of Theorem csbexa
StepHypRef Expression
1 csbexa.1 . . 3  |-  A  e. 
_V
2 csbexga 3967 . . 3  |-  ( ( A  e.  _V  /\  A. x  B  e.  _V )  ->  [_ A  /  x ]_ B  e.  _V )
31, 2mpan 415 . 2  |-  ( A. x  B  e.  _V  ->  [_ A  /  x ]_ B  e.  _V )
4 csbexa.2 . 2  |-  B  e. 
_V
53, 4mpg 1385 1  |-  [_ A  /  x ]_ B  e. 
_V
Colors of variables: wff set class
Syntax hints:   A.wal 1287    e. wcel 1438   _Vcvv 2619   [_csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by:  dfmpt2  5988
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