Theorem indifdir

Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011) (Revised by BTernaryTau, 14-Aug-2024)

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

Proof

Step	Hyp	Ref	Expression
1		indifdi	$$⊢C\cap \left(A\setminus B\right)=\left(C\cap A\right)\setminus \left(C\cap B\right)$$
2		incom	$$⊢\left(A\setminus B\right)\cap C=C\cap \left(A\setminus B\right)$$
3		incom	$$⊢A\cap C=C\cap A$$
4		incom	$$⊢B\cap C=C\cap B$$
5	3 4	difeq12i	$$⊢\left(A\cap C\right)\setminus \left(B\cap C\right)=\left(C\cap A\right)\setminus \left(C\cap B\right)$$
6	1 2 5	3eqtr4i	$$⊢\left(A\setminus B\right)\cap C=\left(A\cap C\right)\setminus \left(B\cap C\right)$$