Theorem undir 4207

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 4205	. 2 ⊢ (𝐶 ∪ (𝐴 ∩ 𝐵)) = ((𝐶 ∪ 𝐴) ∩ (𝐶 ∪ 𝐵))
2		uncom 4083	. 2 ⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐶 ∪ (𝐴 ∩ 𝐵))
3		uncom 4083	. . 3 ⊢ (𝐴 ∪ 𝐶) = (𝐶 ∪ 𝐴)
4		uncom 4083	. . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
5	3, 4	ineq12i 4141	. 2 ⊢ ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶)) = ((𝐶 ∪ 𝐴) ∩ (𝐶 ∪ 𝐵))
6	1, 2, 5	3eqtr4i 2776	1 ⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶))

Syntax hints:   = wceq 1539   ∪ cun 3881   ∩ cin 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-un 3888  df-in 3890

This theorem is referenced by:  undif1  4406  dfif4  4471  dfif5  4472  bwth  22469