Theorem indi 3502

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 837	. . . 4 |- ((x e. A /\ (x e. B \/ x e. C)) <-> ((x e. A /\ x e. B) \/ (x e. A /\ x e. C)))
2		elin 3220	. . . . 5 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
3		elin 3220	. . . . 5 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
4	2, 3	orbi12i 507	. . . 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 243	. . 3 |- ((x e. A /\ (x e. B \/ x e. C)) <-> (x e. (A i^i B) \/ x e. (A i^i C)))
6		elun 3221	. . . 4 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
7	6	anbi2i 675	. . 3 |- ((x e. A /\ x e. (B u. C)) <-> (x e. A /\ (x e. B \/ x e. C)))
8		elun 3221	. . 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 268	. 2 |- ((x e. A /\ x e. (B u. C)) <-> x e. ((A i^i B) u. (A i^i C)))
10	9	ineqri 3450	1 |- (A i^i (B u. C)) = ((A i^i B) u. (A i^i C))

Syntax hints:   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710   u. cun 3208   i^i cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215

This theorem is referenced by:  indir  3504  difindi  3510  undisj2  3604  disjssun  3609  difdifdir  3638  diftpsn3  3850  addcass  4416  resundi  4982  sbthlem1  6204