Theorem undi 3874

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 3796	. . . 4 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
2	1	orbi2i 541	. . 3 |- ( ( x e. A \/ x e. ( B i^i C ) ) <-> ( x e. A \/ ( x e. B /\ x e. C ) ) )
3		ordi 908	. . 3 |- ( ( x e. A \/ ( x e. B /\ x e. C ) ) <-> ( ( x e. A \/ x e. B ) /\ ( x e. A \/ x e. C ) ) )
4		elin 3796	. . . 4 |- ( x e. ( ( A u. B ) i^i ( A u. C ) ) <-> ( x e. ( A u. B ) /\ x e. ( A u. C ) ) )
5		elun 3753	. . . . 5 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6		elun 3753	. . . . 5 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
7	5, 6	anbi12i 733	. . . 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 265	. . 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 286	. 2 |- ( ( x e. A \/ x e. ( B i^i C ) ) <-> x e. ( ( A u. B ) i^i ( A u. C ) ) )
10	9	uneqri 3755	1 |- ( A u. ( B i^i C ) ) = ( ( A u. B ) i^i ( A u. C ) )

Syntax hints:   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581

This theorem is referenced by:  undir  3876  dfif4  4101  dfif5  4102