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 ∩ ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 ∩ 𝐵 ) ∖ 𝐶 )

Proof

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