Theorem undi 3212

Description: Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
undi	|- ( A u. ( B i^i C ) ) = ( ( A u. B ) i^i ( A u. C ) )

Proof of Theorem undi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3155	. . . 4 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
2	1	orbi2i 711	. . 3 |- ( ( x e. A \/ x e. ( B i^i C ) ) <-> ( x e. A \/ ( x e. B /\ x e. C ) ) )
3		ordi 762	. . 3 |- ( ( x e. A \/ ( x e. B /\ x e. C ) ) <-> ( ( x e. A \/ x e. B ) /\ ( x e. A \/ x e. C ) ) )
4		elin 3155	. . . 4 |- ( x e. ( ( A u. B ) i^i ( A u. C ) ) <-> ( x e. ( A u. B ) /\ x e. ( A u. C ) ) )
5		elun 3113	. . . . 5 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6		elun 3113	. . . . 5 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
7	5, 6	anbi12i 447	. . . 4 |- ( ( x e. ( A u. B ) /\ x e. ( A u. C ) ) <-> ( ( x e. A \/ x e. B ) /\ ( x e. A \/ x e. C ) ) )
8	4, 7	bitr2i 183	. . 3 |- ( ( ( x e. A \/ x e. B ) /\ ( x e. A \/ x e. C ) ) <-> x e. ( ( A u. B ) i^i ( A u. C ) ) )
9	2, 3, 8	3bitri 204	. 2 |- ( ( x e. A \/ x e. ( B i^i C ) ) <-> x e. ( ( A u. B ) i^i ( A u. C ) ) )
10	9	uneqri 3114	1 |- ( A u. ( B i^i C ) ) = ( ( A u. B ) i^i ( A u. C ) )

Syntax hints:   /\ wa 102   \/ wo 661   = wceq 1284   e. wcel 1433   u. cun 2971   i^i cin 2972

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979

This theorem is referenced by:  undir  3214