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Theorem csbprc 3830
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 3431 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 sbcex 3337 . . . . . . 7  |-  ( [. A  /  x ]. y  e.  B  ->  A  e. 
_V )
32con3i 135 . . . . . 6  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. y  e.  B
)
43pm2.21d 106 . . . . 5  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. y  e.  B  -> F.  ) )
5 falim 1409 . . . . 5  |-  ( F. 
->  [. A  /  x ]. y  e.  B
)
64, 5impbid1 203 . . . 4  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. y  e.  B  <-> F.  ) )
76abbidv 2593 . . 3  |-  ( -.  A  e.  _V  ->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  | F.  }
)
8 fal 1402 . . . 4  |-  -. F.
98abf 3828 . . 3  |-  { y  | F.  }  =  (/)
107, 9syl6eq 2514 . 2  |-  ( -.  A  e.  _V  ->  { y  |  [. A  /  x ]. y  e.  B }  =  (/) )
111, 10syl5eq 2510 1  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1395   F. wfal 1400    e. wcel 1819   {cab 2442   _Vcvv 3109   [.wsbc 3327   [_csb 3430   (/)c0 3793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794
This theorem is referenced by:  csb0  3831  sbcel12  3832  sbcne12  3836  sbcel2  3839  csbidm  3853  csbun  3862  csbin  3863  csbif  3994  csbuni  4279  sbcbr123  4507  sbcbr  4509  csbexg  4589  csbopab  4788  csbxp  5090  csbres  5286  csbima12  5364  csbrn  5474  csbiota  5586  csbfv12  5907  csbfv  5909  csbriota  6270  csbov123  6330  csbov  6331
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