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Theorem csbexg 4579
Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 17-Aug-2018.)
Assertion
Ref Expression
csbexg  |-  ( 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 3436 . . 3  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 abid2 2607 . . . . . . . 8  |-  { y  |  y  e.  B }  =  B
3 elex 3122 . . . . . . . 8  |-  ( B  e.  W  ->  B  e.  _V )
42, 3syl5eqel 2559 . . . . . . 7  |-  ( B  e.  W  ->  { y  |  y  e.  B }  e.  _V )
54alimi 1614 . . . . . 6  |-  ( A. x  B  e.  W  ->  A. x { y  |  y  e.  B }  e.  _V )
6 spsbc 3344 . . . . . 6  |-  ( A  e.  _V  ->  ( A. x { y  |  y  e.  B }  e.  _V  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V ) )
75, 6syl5 32 . . . . 5  |-  ( A  e.  _V  ->  ( A. x  B  e.  W  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V ) )
87imp 429 . . . 4  |-  ( ( A  e.  _V  /\  A. x  B  e.  W
)  ->  [. A  /  x ]. { y  |  y  e.  B }  e.  _V )
9 nfcv 2629 . . . . . 6  |-  F/_ x _V
109sbcabel 3420 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. { y  |  y  e.  B }  e.  _V 
<->  { y  |  [. A  /  x ]. y  e.  B }  e.  _V ) )
1110adantr 465 . . . 4  |-  ( ( 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 210 . . 3  |-  ( ( A  e.  _V  /\  A. x  B  e.  W
)  ->  { y  |  [. A  /  x ]. y  e.  B }  e.  _V )
131, 12syl5eqel 2559 . 2  |-  ( ( A  e.  _V  /\  A. x  B  e.  W
)  ->  [_ A  /  x ]_ B  e.  _V )
14 csbprc 3821 . . . 4  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
15 0ex 4577 . . . 4  |-  (/)  e.  _V
1614, 15syl6eqel 2563 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  e.  _V )
1716adantr 465 . 2  |-  ( ( -.  A  e.  _V  /\ 
A. x  B  e.  W )  ->  [_ A  /  x ]_ B  e. 
_V )
1813, 17pm2.61ian 788 1  |-  ( A. x  B  e.  W  ->  [_ A  /  x ]_ B  e.  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    e. wcel 1767   {cab 2452   _Vcvv 3113   [.wsbc 3331   [_csb 3435   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786
This theorem is referenced by:  csbex  4580  abfmpeld  27261
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