Theorem undir 4287

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 4285	. 2 ⊢ (𝐶 ∪ (𝐴 ∩ 𝐵)) = ((𝐶 ∪ 𝐴) ∩ (𝐶 ∪ 𝐵))
2		uncom 4158	. 2 ⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐶 ∪ (𝐴 ∩ 𝐵))
3		uncom 4158	. . 3 ⊢ (𝐴 ∪ 𝐶) = (𝐶 ∪ 𝐴)
4		uncom 4158	. . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵)
5	3, 4	ineq12i 4218	. 2 ⊢ ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶)) = ((𝐶 ∪ 𝐴) ∩ (𝐶 ∪ 𝐵))
6	1, 2, 5	3eqtr4i 2775	1 ⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶))

Syntax hints:   = wceq 1540   ∪ cun 3949   ∩ cin 3950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-un 3956  df-in 3958

This theorem is referenced by:  undif1  4476  dfif4  4541  dfif5  4542  bwth  23418