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Theorem csbsng 3422
Description: Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbsng  V  [_  ]_ { }  { [_  ]_ }

Proof of Theorem csbsng
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 csbabg 2901 . . 3  V  [_  ]_ {  |  }  {  |  [.  ].  }
2 sbceq2g 2866 . . . 4  V  [.  ]. 
[_  ]_
32abbidv 2152 . . 3  V  {  |  [.  ].  }  {  |  [_  ]_ }
41, 3eqtrd 2069 . 2  V  [_  ]_ {  |  }  {  |  [_  ]_ }
5 df-sn 3373 . . 3  { }  {  |  }
65csbeq2i 2870 . 2  [_  ]_ { }  [_  ]_ {  |  }
7 df-sn 3373 . 2  { [_  ]_ }  {  |  [_  ]_ }
84, 6, 73eqtr4g 2094 1  V  [_  ]_ { }  { [_  ]_ }
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   wcel 1390   {cab 2023   [.wsbc 2758   [_csb 2846   {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847  df-sn 3373
This theorem is referenced by: (None)
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