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Theorem csbunig 3714
Description: Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbunig  |-  ( A  e.  V  ->  [_ A  /  x ]_ U. B  =  U. [_ A  /  x ]_ B )

Proof of Theorem csbunig
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 3031 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { z  |  E. y ( z  e.  y  /\  y  e.  B ) }  =  { z  |  [. A  /  x ]. E. y ( z  e.  y  /\  y  e.  B ) } )
2 sbcexg 2935 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. y ( z  e.  y  /\  y  e.  B )  <->  E. y [. A  /  x ]. ( z  e.  y  /\  y  e.  B
) ) )
3 sbcang 2924 . . . . . . 7  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( z  e.  y  /\  y  e.  B
)  <->  ( [. A  /  x ]. z  e.  y  /\  [. A  /  x ]. y  e.  B ) ) )
4 sbcg 2950 . . . . . . . 8  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  y  <->  z  e.  y ) )
5 sbcel2g 2994 . . . . . . . 8  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  [_ A  /  x ]_ B ) )
64, 5anbi12d 464 . . . . . . 7  |-  ( A  e.  V  ->  (
( [. A  /  x ]. z  e.  y  /\  [. A  /  x ]. y  e.  B
)  <->  ( z  e.  y  /\  y  e. 
[_ A  /  x ]_ B ) ) )
73, 6bitrd 187 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( z  e.  y  /\  y  e.  B
)  <->  ( z  e.  y  /\  y  e. 
[_ A  /  x ]_ B ) ) )
87exbidv 1781 . . . . 5  |-  ( A  e.  V  ->  ( E. y [. A  /  x ]. ( z  e.  y  /\  y  e.  B )  <->  E. y
( z  e.  y  /\  y  e.  [_ A  /  x ]_ B
) ) )
92, 8bitrd 187 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. y ( z  e.  y  /\  y  e.  B )  <->  E. y
( z  e.  y  /\  y  e.  [_ A  /  x ]_ B
) ) )
109abbidv 2235 . . 3  |-  ( A  e.  V  ->  { z  |  [. A  /  x ]. E. y ( z  e.  y  /\  y  e.  B ) }  =  { z  |  E. y ( z  e.  y  /\  y  e.  [_ A  /  x ]_ B ) } )
111, 10eqtrd 2150 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { z  |  E. y ( z  e.  y  /\  y  e.  B ) }  =  { z  |  E. y ( z  e.  y  /\  y  e.  [_ A  /  x ]_ B ) } )
12 df-uni 3707 . . 3  |-  U. B  =  { z  |  E. y ( z  e.  y  /\  y  e.  B ) }
1312csbeq2i 2999 . 2  |-  [_ A  /  x ]_ U. B  =  [_ A  /  x ]_ { z  |  E. y ( z  e.  y  /\  y  e.  B ) }
14 df-uni 3707 . 2  |-  U. [_ A  /  x ]_ B  =  { z  |  E. y ( z  e.  y  /\  y  e. 
[_ A  /  x ]_ B ) }
1511, 13, 143eqtr4g 2175 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ U. B  =  U. [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103   [.wsbc 2882   [_csb 2975   U.cuni 3706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976  df-uni 3707
This theorem is referenced by: (None)
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