Theorem inindi

Description: Intersection distributes over itself. (Contributed by NM, 6-May-1994)

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

Proof

Step	Hyp	Ref	Expression
1		inidm	$$⊢A\cap A=A$$
2	1	ineq1i	$$⊢\left(A\cap A\right)\cap \left(B\cap C\right)=A\cap \left(B\cap C\right)$$
3		in4	$$⊢\left(A\cap A\right)\cap \left(B\cap C\right)=\left(A\cap B\right)\cap \left(A\cap C\right)$$
4	2 3	eqtr3i	$$⊢A\cap \left(B\cap C\right)=\left(A\cap B\right)\cap \left(A\cap C\right)$$