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Theorem csbabg 2901
Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
csbabg  V  [_  ]_ {  |  }  {  |  [.  ]. }
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   ()    V(,)

Proof of Theorem csbabg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbccom 2827 . . . 4  [.  ]. [.  ].  [.  ]. [.  ].
2 df-clab 2024 . . . . 5  {  | 
[.  ]. }  [.  ].
3 sbsbc 2762 . . . . 5  [.  ].  [.  ]. [.  ].
42, 3bitri 173 . . . 4  {  | 
[.  ]. }  [.  ]. [.  ].
5 df-clab 2024 . . . . . 6  {  |  }
6 sbsbc 2762 . . . . . 6  [.  ].
75, 6bitri 173 . . . . 5  {  |  }  [.  ].
87sbcbii 2812 . . . 4  [.  ].  {  |  } 
[.  ].
[.  ].
91, 4, 83bitr4i 201 . . 3  {  | 
[.  ]. }  [.  ].  {  |  }
10 sbcel2g 2865 . . 3  V  [.  ].  {  |  } 
[_  ]_ {  |  }
119, 10syl5rbb 182 . 2  V  [_  ]_ {  |  }  {  |  [.  ]. }
1211eqrdv 2035 1  V  [_  ]_ {  |  }  {  |  [.  ]. }
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   wcel 1390  wsb 1642   {cab 2023   [.wsbc 2758   [_csb 2846
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-bnd 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
This theorem is referenced by:  csbsng  3422  csbunig  3579  csbxpg  4364  csbdmg  4472  csbrng  4725
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