Theorem undir 4239

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 4237	. 2 ⊢ (𝐶 ∪ (𝐴 ∩ 𝐵)) = ((𝐶 ∪ 𝐴) ∩ (𝐶 ∪ 𝐵))
2		uncom 4111	. 2 ⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐶 ∪ (𝐴 ∩ 𝐵))
3		uncom 4111	. . 3 ⊢ (𝐴 ∪ 𝐶) = (𝐶 ∪ 𝐴)
4		uncom 4111	. . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
5	3, 4	ineq12i 4170	. 2 ⊢ ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶)) = ((𝐶 ∪ 𝐴) ∩ (𝐶 ∪ 𝐵))
6	1, 2, 5	3eqtr4i 2795	1 ⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶))

Syntax hints:   = wceq 1560   ∪ cun 3902   ∩ cin 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-un 3909  df-in 3911

This theorem is referenced by:  undif1  4430  dfif4  4496  dfif5  4497  bwth  23467