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 ∪ B) ∩ C) = ((A ∩ C) ∪ (B ∩ C))

Proof of Theorem indir

Step	Hyp	Ref	Expression
1		indi 3502	. 2 ⊢ (C ∩ (A ∪ B)) = ((C ∩ A) ∪ (C ∩ B))
2		incom 3449	. 2 ⊢ ((A ∪ B) ∩ C) = (C ∩ (A ∪ B))
3		incom 3449	. . 3 ⊢ (A ∩ C) = (C ∩ A)
4		incom 3449	. . 3 ⊢ (B ∩ C) = (C ∩ B)
5	3, 4	uneq12i 3417	. 2 ⊢ ((A ∩ C) ∪ (B ∩ C)) = ((C ∩ A) ∪ (C ∩ B))
6	1, 2, 5	3eqtr4i 2383	1 ⊢ ((A ∪ B) ∩ C) = ((A ∩ C) ∪ (B ∩ C))

Syntax hints:   = wceq 1642   ∪ cun 3208   ∩ 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