Theorem in12

Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001)

	Ref	Expression
Assertion	in12	$$⊢A\cap \left(B\cap C\right)=B\cap \left(A\cap C\right)$$

Proof

Step	Hyp	Ref	Expression
1		incom	$$⊢A\cap B=B\cap A$$
2	1	ineq1i	$$⊢\left(A\cap B\right)\cap C=\left(B\cap A\right)\cap C$$
3		inass	$$⊢\left(A\cap B\right)\cap C=A\cap \left(B\cap C\right)$$
4		inass	$$⊢\left(B\cap A\right)\cap C=B\cap \left(A\cap C\right)$$
5	2 3 4	3eqtr3i	$$⊢A\cap \left(B\cap C\right)=B\cap \left(A\cap C\right)$$