Theorem difindir

Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004)

	Ref	Expression
Assertion	difindir	⊢ ( ( 𝐴 ∩ 𝐵 ) ∖ 𝐶 ) = ( ( 𝐴 ∖ 𝐶 ) ∩ ( 𝐵 ∖ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		inindir	⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) = ( ( 𝐴 ∩ ( V ∖ 𝐶 ) ) ∩ ( 𝐵 ∩ ( V ∖ 𝐶 ) ) )
2		invdif	⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ ( V ∖ 𝐶 ) ) = ( ( 𝐴 ∩ 𝐵 ) ∖ 𝐶 )
3		invdif	⊢ ( 𝐴 ∩ ( V ∖ 𝐶 ) ) = ( 𝐴 ∖ 𝐶 )
4		invdif	⊢ ( 𝐵 ∩ ( V ∖ 𝐶 ) ) = ( 𝐵 ∖ 𝐶 )
5	3 4	ineq12i	⊢ ( ( 𝐴 ∩ ( V ∖ 𝐶 ) ) ∩ ( 𝐵 ∩ ( V ∖ 𝐶 ) ) ) = ( ( 𝐴 ∖ 𝐶 ) ∩ ( 𝐵 ∖ 𝐶 ) )
6	1 2 5	3eqtr3i	⊢ ( ( 𝐴 ∩ 𝐵 ) ∖ 𝐶 ) = ( ( 𝐴 ∖ 𝐶 ) ∩ ( 𝐵 ∖ 𝐶 ) )