Theorem indir

Description: Distributive law for intersection over union. Theorem 28 of Suppes p. 27. (Contributed by NM, 30-Sep-2002)

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

Proof

Step	Hyp	Ref	Expression
1		indi	$$⊢C\cap \left(A\cup B\right)=\left(C\cap A\right)\cup \left(C\cap B\right)$$
2		incom	$$⊢\left(A\cup B\right)\cap C=C\cap \left(A\cup B\right)$$
3		incom	$$⊢A\cap C=C\cap A$$
4		incom	$$⊢B\cap C=C\cap B$$
5	3 4	uneq12i	$$⊢\left(A\cap C\right)\cup \left(B\cap C\right)=\left(C\cap A\right)\cup \left(C\cap B\right)$$
6	1 2 5	3eqtr4i	$$⊢\left(A\cup B\right)\cap C=\left(A\cap C\right)\cup \left(B\cap C\right)$$