Theorem indir 3504

Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Assertion

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

Proof of Theorem indir

Step	Hyp	Ref	Expression
1		indi 3502	. 2 |- (C i^i (A u. B)) = ((C i^i A) u. (C i^i B))
2		incom 3449	. 2 |- ((A u. B) i^i C) = (C i^i (A u. B))
3		incom 3449	. . 3 |- (A i^i C) = (C i^i A)
4		incom 3449	. . 3 |- (B i^i C) = (C i^i B)
5	3, 4	uneq12i 3417	. 2 |- ((A i^i C) u. (B i^i C)) = ((C i^i A) u. (C i^i B))
6	1, 2, 5	3eqtr4i 2383	1 |- ((A u. B) i^i C) = ((A i^i C) u. (B i^i C))

Syntax hints:   = wceq 1642   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:  difundir  3509  undisj1  3603  addcass  4416  nnsucelrlem3  4427  resundir  4983