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Theorem csbexga 3926
Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbexga  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  [_ A  /  x ]_ B  e.  _V )

Proof of Theorem csbexga
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-csb 2918 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 abid2 2203 . . . . . . 7  |-  { y  |  y  e.  B }  =  B
3 elex 2619 . . . . . . 7  |-  ( B  e.  W  ->  B  e.  _V )
42, 3syl5eqel 2169 . . . . . 6  |-  ( B  e.  W  ->  { y  |  y  e.  B }  e.  _V )
54alimi 1385 . . . . 5  |-  ( A. x  B  e.  W  ->  A. x { y  |  y  e.  B }  e.  _V )
6 spsbc 2835 . . . . 5  |-  ( A  e.  V  ->  ( A. x { y  |  y  e.  B }  e.  _V  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V ) )
75, 6syl5 32 . . . 4  |-  ( A  e.  V  ->  ( A. x  B  e.  W  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V ) )
87imp 122 . . 3  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V )
9 nfcv 2223 . . . . 5  |-  F/_ x _V
109sbcabel 2904 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  |  y  e.  B }  e.  _V 
<->  { y  |  [. A  /  x ]. y  e.  B }  e.  _V ) )
1110adantr 270 . . 3  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  ( [. A  /  x ]. {
y  |  y  e.  B }  e.  _V  <->  { y  |  [. A  /  x ]. y  e.  B }  e.  _V ) )
128, 11mpbid 145 . 2  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  { y  |  [. A  /  x ]. y  e.  B }  e.  _V )
131, 12syl5eqel 2169 1  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  [_ A  /  x ]_ B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    e. wcel 1434   {cab 2069   _Vcvv 2610   [.wsbc 2824   [_csb 2917
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825  df-csb 2918
This theorem is referenced by:  csbexa  3927
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