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Theorem csbunig 3579
Description: Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbunig  V  [_  ]_ U.  U. [_  ]_

Proof of Theorem csbunig
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 2901 . . 3  V  [_  ]_ {  |  }  {  |  [.  ].  }
2 sbcexg 2807 . . . . 5  V  [.  ].  [.  ].
3 sbcang 2800 . . . . . . 7  V  [.  ].  [.  ].  [.  ].
4 sbcg 2821 . . . . . . . 8  V  [.  ].
5 sbcel2g 2865 . . . . . . . 8  V  [.  ]. 
[_  ]_
64, 5anbi12d 442 . . . . . . 7  V  [.  ].  [.  ].  [_  ]_
73, 6bitrd 177 . . . . . 6  V  [.  ].  [_  ]_
87exbidv 1703 . . . . 5  V  [.  ].  [_  ]_
92, 8bitrd 177 . . . 4  V  [.  ].  [_  ]_
109abbidv 2152 . . 3  V  {  |  [.  ].  }  {  |  [_  ]_ }
111, 10eqtrd 2069 . 2  V  [_  ]_ {  |  }  {  |  [_  ]_ }
12 df-uni 3572 . . 3  U.  {  |  }
1312csbeq2i 2870 . 2  [_  ]_ U.  [_  ]_ {  |  }
14 df-uni 3572 . 2  U. [_  ]_  {  |  [_  ]_ }
1511, 13, 143eqtr4g 2094 1  V  [_  ]_ U.  U. [_  ]_
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390   {cab 2023   [.wsbc 2758   [_csb 2846   U.cuni 3571
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  df-uni 3572
This theorem is referenced by: (None)
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