Theorem indif1

Description: Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015)

	Ref	Expression
Assertion	indif1	⊢ ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ( ( 𝐴 ∩ 𝐵 ) ∖ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		indif2	⊢ ( 𝐵 ∩ ( 𝐴 ∖ 𝐶 ) ) = ( ( 𝐵 ∩ 𝐴 ) ∖ 𝐶 )
2		incom	⊢ ( 𝐵 ∩ ( 𝐴 ∖ 𝐶 ) ) = ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 )
3		incom	⊢ ( 𝐵 ∩ 𝐴 ) = ( 𝐴 ∩ 𝐵 )
4	3	difeq1i	⊢ ( ( 𝐵 ∩ 𝐴 ) ∖ 𝐶 ) = ( ( 𝐴 ∩ 𝐵 ) ∖ 𝐶 )
5	1 2 4	3eqtr3i	⊢ ( ( 𝐴 ∖ 𝐶 ) ∩ 𝐵 ) = ( ( 𝐴 ∩ 𝐵 ) ∖ 𝐶 )