Theorem difindir

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

	Ref	Expression
Assertion	difindir	|- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) )

Proof

Step	Hyp	Ref	Expression
1		inindir	|- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) )
2		invdif	|- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i B ) \ C )
3		invdif	|- ( A i^i ( _V \ C ) ) = ( A \ C )
4		invdif	|- ( B i^i ( _V \ C ) ) = ( B \ C )
5	3 4	ineq12i	|- ( ( A i^i ( _V \ C ) ) i^i ( B i^i ( _V \ C ) ) ) = ( ( A \ C ) i^i ( B \ C ) )
6	1 2 5	3eqtr3i	|- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) )