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Theorem csbunig 3644
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 2978 . . 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 2882 . . . . 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 2871 . . . . . . 7  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( z  e.  y  /\  y  e.  B
)  <->  ( [. A  /  x ]. z  e.  y  /\  [. A  /  x ]. y  e.  B ) ) )
4 sbcg 2897 . . . . . . . 8  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  y  <->  z  e.  y ) )
5 sbcel2g 2941 . . . . . . . 8  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  [_ A  /  x ]_ B ) )
64, 5anbi12d 457 . . . . . . 7  |-  ( A  e.  V  ->  (
( [. A  /  x ]. z  e.  y  /\  [. A  /  x ]. y  e.  B
)  <->  ( z  e.  y  /\  y  e. 
[_ A  /  x ]_ B ) ) )
73, 6bitrd 186 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( z  e.  y  /\  y  e.  B
)  <->  ( z  e.  y  /\  y  e. 
[_ A  /  x ]_ B ) ) )
87exbidv 1750 . . . . 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 186 . . . 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 2202 . . 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 2117 . 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 3637 . . 3  |-  U. B  =  { z  |  E. y ( z  e.  y  /\  y  e.  B ) }
1312csbeq2i 2946 . 2  |-  [_ A  /  x ]_ U. B  =  [_ A  /  x ]_ { z  |  E. y ( z  e.  y  /\  y  e.  B ) }
14 df-uni 3637 . 2  |-  U. [_ A  /  x ]_ B  =  { z  |  E. y ( z  e.  y  /\  y  e. 
[_ A  /  x ]_ B ) }
1511, 13, 143eqtr4g 2142 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ U. B  =  U. [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287   E.wex 1424    e. wcel 1436   {cab 2071   [.wsbc 2829   [_csb 2922   U.cuni 3636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-sbc 2830  df-csb 2923  df-uni 3637
This theorem is referenced by: (None)
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