Theorem undir 3457

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 3455	. 2 ⊢ (𝐶 ∪ (𝐴 ∩ 𝐵)) = ((𝐶 ∪ 𝐴) ∩ (𝐶 ∪ 𝐵))
2		uncom 3351	. 2 ⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐶 ∪ (𝐴 ∩ 𝐵))
3		uncom 3351	. . 3 ⊢ (𝐴 ∪ 𝐶) = (𝐶 ∪ 𝐴)
4		uncom 3351	. . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
5	3, 4	ineq12i 3406	. 2 ⊢ ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶)) = ((𝐶 ∪ 𝐴) ∩ (𝐶 ∪ 𝐵))
6	1, 2, 5	3eqtr4i 2262	1 ⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶))

Syntax hints:   = wceq 1397   ∪ cun 3198   ∩ cin 3199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206

This theorem is referenced by: (None)