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Theorem csbexg 3125
Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbexg  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  [_ A  /  x ]_ B  e.  _V )

Proof of Theorem csbexg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-csb 3116 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 abid2 2433 . . . . . . 7  |-  { y  |  y  e.  B }  =  B
3 elex 2830 . . . . . . 7  |-  ( B  e.  W  ->  B  e.  _V )
42, 3syl5eqel 2400 . . . . . 6  |-  ( B  e.  W  ->  { y  |  y  e.  B }  e.  _V )
54alimi 1550 . . . . 5  |-  ( A. x  B  e.  W  ->  A. x { y  |  y  e.  B }  e.  _V )
6 spsbc 3037 . . . . 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 418 . . 3  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V )
9 nfcv 2452 . . . . 5  |-  F/_ x _V
109sbcabel 3102 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  |  y  e.  B }  e.  _V 
<->  { y  |  [. A  /  x ]. y  e.  B }  e.  _V ) )
1110adantr 451 . . 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 201 . 2  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  { y  |  [. A  /  x ]. y  e.  B }  e.  _V )
131, 12syl5eqel 2400 1  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  [_ A  /  x ]_ B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1531    e. wcel 1701   {cab 2302   _Vcvv 2822   [.wsbc 3025   [_csb 3115
This theorem is referenced by:  csbex  3126  issubc  13761  itgparts  19447  abfmpeld  23215  abfmpel  23216  unirep  25498  cdlemk40  30924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-sbc 3026  df-csb 3116
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