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Theorem csbdmg 4472
Description: Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)
Assertion
Ref Expression
csbdmg  V  [_  ]_ dom  dom  [_  ]_

Proof of Theorem csbdmg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 2901 . . 3  V  [_  ]_ {  |  <. ,  >.  }  {  |  [.  ]. <. ,  >.  }
2 sbcex2 2806 . . . . 5  [.  ]. <. ,  >.  [.  ]. <. ,  >.
3 sbcel2g 2865 . . . . . 6  V  [.  ].
<. ,  >.  <. ,  >.  [_  ]_
43exbidv 1703 . . . . 5  V  [.  ]. <. ,  >.  <. ,  >. 
[_  ]_
52, 4syl5bb 181 . . . 4  V  [.  ]. <. ,  >.  <. ,  >. 
[_  ]_
65abbidv 2152 . . 3  V  {  |  [.  ]. <. ,  >.  }  {  |  <. ,  >.  [_  ]_ }
71, 6eqtrd 2069 . 2  V  [_  ]_ {  |  <. ,  >.  }  {  |  <. ,  >.  [_  ]_ }
8 dfdm3 4465 . . 3  dom  {  |  <. ,  >.  }
98csbeq2i 2870 . 2  [_  ]_ dom  [_  ]_ {  |  <. ,  >.  }
10 dfdm3 4465 . 2  dom  [_  ]_  {  |  <. ,  >.  [_  ]_ }
117, 9, 103eqtr4g 2094 1  V  [_  ]_ dom  dom  [_  ]_
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242  wex 1378   wcel 1390   {cab 2023   [.wsbc 2758   [_csb 2846   <.cop 3370   dom cdm 4288
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-br 3756  df-dm 4298
This theorem is referenced by:  sbcfng  4987
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