Theorem unundi 4163

Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
unundi	⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶))

Proof of Theorem unundi

Step	Hyp	Ref	Expression
1		unidm 4145	. . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴
2	1	uneq1i 4152	. 2 ⊢ ((𝐴 ∪ 𝐴) ∪ (𝐵 ∪ 𝐶)) = (𝐴 ∪ (𝐵 ∪ 𝐶))
3		un4 4162	. 2 ⊢ ((𝐴 ∪ 𝐴) ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶))
4	2, 3	eqtr3i 2754	1 ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶))

Syntax hints:   = wceq 1533   ∪ cun 3939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3946

This theorem is referenced by:  dfif5  4537