Theorem undir

Description: Distributive law for union over intersection. Theorem 29 of Suppes p. 27. (Contributed by NM, 30-Sep-2002)

	Ref	Expression
Assertion	undir	$$⊢\left(A\cap B\right)\cup C=\left(A\cup C\right)\cap \left(B\cup C\right)$$

Proof

Step	Hyp	Ref	Expression
1		undi	$$⊢C\cup \left(A\cap B\right)=\left(C\cup A\right)\cap \left(C\cup B\right)$$
2		uncom	$$⊢\left(A\cap B\right)\cup C=C\cup \left(A\cap B\right)$$
3		uncom	$$⊢A\cup C=C\cup A$$
4		uncom	$$⊢B\cup C=C\cup B$$
5	3 4	ineq12i	$$⊢\left(A\cup C\right)\cap \left(B\cup C\right)=\left(C\cup A\right)\cap \left(C\cup B\right)$$
6	1 2 5	3eqtr4i	$$⊢\left(A\cap B\right)\cup C=\left(A\cup C\right)\cap \left(B\cup C\right)$$