Theorem indi 3178

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 730	. . . 4 |- ((x e. A /\ (x e. B \/ x e. C)) <-> ((x e. A /\ x e. B) \/ (x e. A /\ x e. C)))
2		elin 3120	. . . . 5 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
3		elin 3120	. . . . 5 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
4	2, 3	orbi12i 680	. . . 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 176	. . 3 |- ((x e. A /\ (x e. B \/ x e. C)) <-> (x e. (A i^i B) \/ x e. (A i^i C)))
6		elun 3078	. . . 4 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
7	6	anbi2i 430	. . 3 |- ((x e. A /\ x e. (B u. C)) <-> (x e. A /\ (x e. B \/ x e. C)))
8		elun 3078	. . 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 201	. 2 |- ((x e. A /\ x e. (B u. C)) <-> x e. ((A i^i B) u. (A i^i C)))
10	9	ineqri 3124	1 |- (A i^i (B u. C)) = ((A i^i B) u. (A i^i C))

Syntax hints:   /\ wa 97   \/ wo 628   = wceq 1242   e. wcel 1390   u. cun 2909   i^i cin 2910

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918

This theorem is referenced by:  indir  3180  undisj2  3274  disjssun  3279  difdifdirss  3301  disjpr2  3425  diftpsn3  3496  resundi  4568