Theorem indi 3454

Description: Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem indi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		andi 825	. . . 4 |- ( ( x e. A /\ ( x e. B \/ x e. C ) ) <-> ( ( x e. A /\ x e. B ) \/ ( x e. A /\ x e. C ) ) )
2		elin 3390	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
3		elin 3390	. . . . 5 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4	2, 3	orbi12i 771	. . . 4 |- ( ( x e. ( A i^i B ) \/ x e. ( A i^i C ) ) <-> ( ( x e. A /\ x e. B ) \/ ( x e. A /\ x e. C ) ) )
5	1, 4	bitr4i 187	. . 3 |- ( ( x e. A /\ ( x e. B \/ x e. C ) ) <-> ( x e. ( A i^i B ) \/ x e. ( A i^i C ) ) )
6		elun 3348	. . . 4 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
7	6	anbi2i 457	. . 3 |- ( ( x e. A /\ x e. ( B u. C ) ) <-> ( x e. A /\ ( x e. B \/ x e. C ) ) )
8		elun 3348	. . 3 |- ( x e. ( ( A i^i B ) u. ( A i^i C ) ) <-> ( x e. ( A i^i B ) \/ x e. ( A i^i C ) ) )
9	5, 7, 8	3bitr4i 212	. 2 |- ( ( x e. A /\ x e. ( B u. C ) ) <-> x e. ( ( A i^i B ) u. ( A i^i C ) ) )
10	9	ineqri 3400	1 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )

Syntax hints:   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2202   u. cun 3198   i^i cin 3199

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206

This theorem is referenced by:  indir  3456  undisj2  3553  disjssun  3558  difdifdirss  3579  disjpr2  3733  diftpsn3  3814  resundi  5026  bitsinv1  12522