Theorem undir 3505

Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
undir	⊢ ((A ∩ B) ∪ C) = ((A ∪ C) ∩ (B ∪ C))

Proof of Theorem undir

Step	Hyp	Ref	Expression
1		undi 3503	. 2 ⊢ (C ∪ (A ∩ B)) = ((C ∪ A) ∩ (C ∪ B))
2		uncom 3409	. 2 ⊢ ((A ∩ B) ∪ C) = (C ∪ (A ∩ B))
3		uncom 3409	. . 3 ⊢ (A ∪ C) = (C ∪ A)
4		uncom 3409	. . 3 ⊢ (B ∪ C) = (C ∪ B)
5	3, 4	ineq12i 3456	. 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:  undif1  3626  dfif4  3674  dfif5  3675  nnsucelrlem3  4427