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Metamath Proof Explorer

Theorem indif2

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

		Ref	Expression
	Assertion	indif2	$⊢ A \cap (B ∖ C) = (A \cap B) ∖ C$

Step	Hyp	Ref	Expression
1		inass	$⊢ (A \cap B) \cap (V ∖ C) = A \cap (B \cap (V ∖ C))$
2		invdif	$⊢ (A \cap B) \cap (V ∖ C) = (A \cap B) ∖ C$
3		invdif	$⊢ B \cap (V ∖ C) = B ∖ C$
4	3	ineq2i	$⊢ A \cap (B \cap (V ∖ C)) = A \cap (B ∖ C)$
5	1 2 4	3eqtr3ri	$⊢ A \cap (B ∖ C) = (A \cap B) ∖ C$