Theorem indif2

Description: Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009)

	Ref	Expression
Assertion	indif2	$$⊢A\cap \left(B\setminus C\right)=\left(A\cap B\right)\setminus C$$

Proof

Step	Hyp	Ref	Expression
1		inass	$$⊢\left(A\cap B\right)\cap \left(\mathrm{V}\setminus C\right)=A\cap \left(B\cap \left(\mathrm{V}\setminus C\right)\right)$$
2		invdif	$$⊢\left(A\cap B\right)\cap \left(\mathrm{V}\setminus C\right)=\left(A\cap B\right)\setminus C$$
3		invdif	$$⊢B\cap \left(\mathrm{V}\setminus C\right)=B\setminus C$$
4	3	ineq2i	$$⊢A\cap \left(B\cap \left(\mathrm{V}\setminus C\right)\right)=A\cap \left(B\setminus C\right)$$
5	1 2 4	3eqtr3ri	$$⊢A\cap \left(B\setminus C\right)=\left(A\cap B\right)\setminus C$$