Theorem coundi 5171

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 4112	. . 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 4084	. . . . . . . 8 |- ( x ( B u. C ) z <-> ( x B z \/ x C z ) )
3	2	anbi1i 458	. . . . . . 7 |- ( ( x ( B u. C ) z /\ z A y ) <-> ( ( x B z \/ x C z ) /\ z A y ) )
4		andir 820	. . . . . . 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 184	. . . . . 6 |- ( ( x ( B u. C ) z /\ z A y ) <-> ( ( x B z /\ z A y ) \/ ( x C z /\ z A y ) ) )
6	5	exbii 1619	. . . . 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 1642	. . . . 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 185	. . . 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 4100	. . 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 2217	. 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 4672	. . 3 |- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) }
12		df-co 4672	. . 3 |- ( A o. C ) = { <. x , y >. | E. z ( x C z /\ z A y ) }
13	11, 12	uneq12i 3315	. 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 4672	. 2 |- ( A o. ( B u. C ) ) = { <. x , y >. | E. z ( x ( B u. C ) z /\ z A y ) }
15	10, 13, 14	3eqtr4ri 2228	1 |- ( A o. ( B u. C ) ) = ( ( A o. B ) u. ( A o. C ) )

Syntax hints:   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1506   u. cun 3155   class class class wbr 4033   {copab 4093   o. ccom 4667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-br 4034  df-opab 4095  df-co 4672

This theorem is referenced by:  relcoi1  5201