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Theorem csbunigOLD 36867
Description: Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 22-Aug-2018. Use csbuni 4250 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbunigOLD  |-  ( A  e.  V  ->  [_ A  /  x ]_ U. B  =  U. [_ A  /  x ]_ B )

Proof of Theorem csbunigOLD
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabgOLD 36866 . . 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 sbcexgOLD 36556 . . . . 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 sbcangOLD 36542 . . . . . . 7  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( z  e.  y  /\  y  e.  B
)  <->  ( [. A  /  x ]. z  e.  y  /\  [. A  /  x ]. y  e.  B ) ) )
4 sbcg 3371 . . . . . . . 8  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  y  <->  z  e.  y ) )
5 sbcel2gOLD 36558 . . . . . . . 8  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  [_ A  /  x ]_ B ) )
64, 5anbi12d 715 . . . . . . 7  |-  ( A  e.  V  ->  (
( [. A  /  x ]. z  e.  y  /\  [. A  /  x ]. y  e.  B
)  <->  ( z  e.  y  /\  y  e. 
[_ A  /  x ]_ B ) ) )
73, 6bitrd 256 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( z  e.  y  /\  y  e.  B
)  <->  ( z  e.  y  /\  y  e. 
[_ A  /  x ]_ B ) ) )
87exbidv 1761 . . . . 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 256 . . . 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 2565 . . 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 2470 . 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 4223 . . 3  |-  U. B  =  { z  |  E. y ( z  e.  y  /\  y  e.  B ) }
1312csbeq2i 3816 . 2  |-  [_ A  /  x ]_ U. B  =  [_ A  /  x ]_ { z  |  E. y ( z  e.  y  /\  y  e.  B ) }
14 df-uni 4223 . 2  |-  U. [_ A  /  x ]_ B  =  { z  |  E. y ( z  e.  y  /\  y  e. 
[_ A  /  x ]_ B ) }
1511, 13, 143eqtr4g 2495 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ U. B  =  U. [_ A  /  x ]_ B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414   [.wsbc 3305   [_csb 3401   U.cuni 4222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-sbc 3306  df-csb 3402  df-uni 4223
This theorem is referenced by:  csbfv12gALTOLD  36868  csbfv12gALTVD  36951
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