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Theorem csbexa 3914
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 3913 . . 3  |-  ( ( A  e.  _V  /\  A. x  B  e.  _V )  ->  [_ A  /  x ]_ B  e.  _V )
31, 2mpan 408 . 2  |-  ( A. x  B  e.  _V  ->  [_ A  /  x ]_ B  e.  _V )
4 csbexa.2 . 2  |-  B  e. 
_V
53, 4mpg 1356 1  |-  [_ A  /  x ]_ B  e. 
_V
Colors of variables: wff set class
Syntax hints:   A.wal 1257    e. wcel 1409   _Vcvv 2574   [_csb 2880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788  df-csb 2881
This theorem is referenced by:  dfmpt2  5872
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