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Theorem csbexga 3885
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 2853 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 abid2 2158 . . . . . . 7  |-  { y  |  y  e.  B }  =  B
3 elex 2566 . . . . . . 7  |-  ( B  e.  W  ->  B  e.  _V )
42, 3syl5eqel 2124 . . . . . 6  |-  ( B  e.  W  ->  { y  |  y  e.  B }  e.  _V )
54alimi 1344 . . . . 5  |-  ( A. x  B  e.  W  ->  A. x { y  |  y  e.  B }  e.  _V )
6 spsbc 2775 . . . . 5  |-  ( A  e.  V  ->  ( A. x { y  |  y  e.  B }  e.  _V  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V ) )
75, 6syl5 28 . . . 4  |-  ( A  e.  V  ->  ( A. x  B  e.  W  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V ) )
87imp 115 . . 3  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V )
9 nfcv 2178 . . . . 5  |-  F/_ x _V
109sbcabel 2839 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  |  y  e.  B }  e.  _V 
<->  { y  |  [. A  /  x ]. y  e.  B }  e.  _V ) )
1110adantr 261 . . 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 135 . 2  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  { y  |  [. A  /  x ]. y  e.  B }  e.  _V )
131, 12syl5eqel 2124 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 97    <-> wb 98   A.wal 1241    e. wcel 1393   {cab 2026   _Vcvv 2557   [.wsbc 2764   [_csb 2852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853
This theorem is referenced by:  csbexa  3886
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