Theorem coundi 5040

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.

Step	Hyp	Ref	Expression
1		unopab 4007	. . 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 3979	. . . . . . . 8 |- ( x ( B u. C ) z <-> ( x B z \/ x C z ) )
3	2	anbi1i 453	. . . . . . 7 |- ( ( x ( B u. C ) z /\ z A y ) <-> ( ( x B z \/ x C z ) /\ z A y ) )
4		andir 808	. . . . . . 7 |- ( ( ( x B z \/ x C z ) /\ z A y ) <-> ( ( x B z /\ z A y ) \/ ( x C z /\ z A y ) ) )
5	3, 4	bitri 183	. . . . . 6 |- ( ( x ( B u. C ) z /\ z A y ) <-> ( ( x B z /\ z A y ) \/ ( x C z /\ z A y ) ) )
6	5	exbii 1584	. . . . 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 1607	. . . . 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 ) ) )
8	6, 7	bitr2i 184	. . . 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 ) )
9	8	opabbii 3995	. . 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 ) }
10	1, 9	eqtri 2160	. 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 4548	. . 3 |- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) }
12		df-co 4548	. . 3 |- ( A o. C ) = { <. x , y >. | E. z ( x C z /\ z A y ) }
13	11, 12	uneq12i 3228	. 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 4548	. 2 |- ( A o. ( B u. C ) ) = { <. x , y >. | E. z ( x ( B u. C ) z /\ z A y ) }
15	10, 13, 14	3eqtr4ri 2171	1 |- ( A o. ( B u. C ) ) = ( ( A o. B ) u. ( A o. C ) )

Syntax hints:   /\ wa 103   \/ wo 697   = wceq 1331   E.wex 1468   u. cun 3069   class class class wbr 3929   {copab 3988   o. ccom 4543

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-br 3930  df-opab 3990  df-co 4548

This theorem is referenced by:  relcoi1  5070