Theorem unundi 4123

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 4104	. . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴
2	1	uneq1i 4111	. 2 ⊢ ((𝐴 ∪ 𝐴) ∪ (𝐵 ∪ 𝐶)) = (𝐴 ∪ (𝐵 ∪ 𝐶))
3		un4 4122	. 2 ⊢ ((𝐴 ∪ 𝐴) ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶))
4	2, 3	eqtr3i 2756	1 ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶))

Syntax hints:   = wceq 1541   ∪ cun 3895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902

This theorem is referenced by:  dfif5  4489