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Theorem csbexg 3261
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 3252 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 abid2 2553 . . . . . . 7  |-  { y  |  y  e.  B }  =  B
3 elex 2964 . . . . . . 7  |-  ( B  e.  W  ->  B  e.  _V )
42, 3syl5eqel 2520 . . . . . 6  |-  ( B  e.  W  ->  { y  |  y  e.  B }  e.  _V )
54alimi 1568 . . . . 5  |-  ( A. x  B  e.  W  ->  A. x { y  |  y  e.  B }  e.  _V )
6 spsbc 3173 . . . . 5  |-  ( A  e.  V  ->  ( A. x { y  |  y  e.  B }  e.  _V  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V ) )
75, 6syl5 30 . . . 4  |-  ( A  e.  V  ->  ( A. x  B  e.  W  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V ) )
87imp 419 . . 3  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V )
9 nfcv 2572 . . . . 5  |-  F/_ x _V
109sbcabel 3238 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  |  y  e.  B }  e.  _V 
<->  { y  |  [. A  /  x ]. y  e.  B }  e.  _V ) )
1110adantr 452 . . 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 202 . 2  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  { y  |  [. A  /  x ]. y  e.  B }  e.  _V )
131, 12syl5eqel 2520 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 177    /\ wa 359   A.wal 1549    e. wcel 1725   {cab 2422   _Vcvv 2956   [.wsbc 3161   [_csb 3251
This theorem is referenced by:  csbex  3262  issubc  14035  itgparts  19931  abfmpeld  24066  abfmpel  24067  unirep  26414  cdlemk40  31714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162  df-csb 3252
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