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 i^i ( B \ C ) ) = ( ( A i^i B ) \ C )

Proof

Step	Hyp	Ref	Expression
1		inass	|- ( ( A i^i B ) i^i ( _V \ C ) ) = ( A i^i ( B i^i ( _V \ C ) ) )
2		invdif	|- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i B ) \ C )
3		invdif	|- ( B i^i ( _V \ C ) ) = ( B \ C )
4	3	ineq2i	|- ( A i^i ( B i^i ( _V \ C ) ) ) = ( A i^i ( B \ C ) )
5	1 2 4	3eqtr3ri	|- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C )