Metamath Proof Explorer


Theorem indif1

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

Ref Expression
Assertion indif1 ( ( 𝐴𝐶 ) ∩ 𝐵 ) = ( ( 𝐴𝐵 ) ∖ 𝐶 )

Proof

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