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Theorem csbprc 3327
Description: The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
csbprc  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )

Proof of Theorem csbprc
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-csb 2934 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 sbcex 2848 . . . . . . 7  |-  ( [. A  /  x ]. y  e.  B  ->  A  e. 
_V )
32con3i 597 . . . . . 6  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. y  e.  B
)
43pm2.21d 584 . . . . 5  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. y  e.  B  -> F.  ) )
5 falim 1303 . . . . 5  |-  ( F. 
->  [. A  /  x ]. y  e.  B
)
64, 5impbid1 140 . . . 4  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. y  e.  B  <-> F.  ) )
76abbidv 2205 . . 3  |-  ( -.  A  e.  _V  ->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  | F.  }
)
8 fal 1296 . . . 4  |-  -. F.
98abf 3325 . . 3  |-  { y  | F.  }  =  (/)
107, 9syl6eq 2136 . 2  |-  ( -.  A  e.  _V  ->  { y  |  [. A  /  x ]. y  e.  B }  =  (/) )
111, 10syl5eq 2132 1  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1289   F. wfal 1294    e. wcel 1438   {cab 2074   _Vcvv 2619   [.wsbc 2840   [_csb 2933   (/)c0 3286
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287
This theorem is referenced by: (None)
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