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Theorem coundi 5450
Description: Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
coundi  |-  ( A  o.  ( B  u.  C ) )  =  ( ( A  o.  B )  u.  ( A  o.  C )
)

Proof of Theorem coundi
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopab 4478 . . 3  |-  ( {
<. x ,  y >.  |  E. z ( x B z  /\  z A y ) }  u.  { <. x ,  y >.  |  E. z ( x C z  /\  z A y ) } )  =  { <. x ,  y >.  |  ( E. z ( x B z  /\  z A y )  \/ 
E. z ( x C z  /\  z A y ) ) }
2 brun 4451 . . . . . . . 8  |-  ( x ( B  u.  C
) z  <->  ( x B z  \/  x C z ) )
32anbi1i 695 . . . . . . 7  |-  ( ( x ( B  u.  C ) z  /\  z A y )  <->  ( (
x B z  \/  x C z )  /\  z A y ) )
4 andir 863 . . . . . . 7  |-  ( ( ( x B z  \/  x C z )  /\  z A y )  <->  ( (
x B z  /\  z A y )  \/  ( x C z  /\  z A y ) ) )
53, 4bitri 249 . . . . . 6  |-  ( ( x ( B  u.  C ) z  /\  z A y )  <->  ( (
x B z  /\  z A y )  \/  ( x C z  /\  z A y ) ) )
65exbii 1635 . . . . 5  |-  ( E. z ( x ( B  u.  C ) z  /\  z A y )  <->  E. z
( ( x B z  /\  z A y )  \/  (
x C z  /\  z A y ) ) )
7 19.43 1661 . . . . 5  |-  ( E. z ( ( x B z  /\  z A y )  \/  ( x C z  /\  z A y ) )  <->  ( E. z ( x B z  /\  z A y )  \/  E. z ( x C z  /\  z A y ) ) )
86, 7bitr2i 250 . . . 4  |-  ( ( E. z ( x B z  /\  z A y )  \/ 
E. z ( x C z  /\  z A y ) )  <->  E. z ( x ( B  u.  C ) z  /\  z A y ) )
98opabbii 4467 . . 3  |-  { <. x ,  y >.  |  ( E. z ( x B z  /\  z A y )  \/ 
E. z ( x C z  /\  z A y ) ) }  =  { <. x ,  y >.  |  E. z ( x ( B  u.  C ) z  /\  z A y ) }
101, 9eqtri 2483 . 2  |-  ( {
<. x ,  y >.  |  E. z ( x B z  /\  z A y ) }  u.  { <. x ,  y >.  |  E. z ( x C z  /\  z A y ) } )  =  { <. x ,  y >.  |  E. z ( x ( B  u.  C ) z  /\  z A y ) }
11 df-co 4960 . . 3  |-  ( A  o.  B )  =  { <. x ,  y
>.  |  E. z
( x B z  /\  z A y ) }
12 df-co 4960 . . 3  |-  ( A  o.  C )  =  { <. x ,  y
>.  |  E. z
( x C z  /\  z A y ) }
1311, 12uneq12i 3619 . 2  |-  ( ( A  o.  B )  u.  ( A  o.  C ) )  =  ( { <. x ,  y >.  |  E. z ( x B z  /\  z A y ) }  u.  {
<. x ,  y >.  |  E. z ( x C z  /\  z A y ) } )
14 df-co 4960 . 2  |-  ( A  o.  ( B  u.  C ) )  =  { <. x ,  y
>.  |  E. z
( x ( B  u.  C ) z  /\  z A y ) }
1510, 13, 143eqtr4ri 2494 1  |-  ( A  o.  ( B  u.  C ) )  =  ( ( A  o.  B )  u.  ( A  o.  C )
)
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1370   E.wex 1587    u. cun 3437   class class class wbr 4403   {copab 4460    o. ccom 4955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3444  df-br 4404  df-opab 4462  df-co 4960
This theorem is referenced by:  relcoi1  5477  mvdco  16074  ustssco  19931  cvmliftlem10  27350  diophren  29323
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