Metamath Proof Explorer


Theorem in13

Description: A rearrangement of intersection. (Contributed by NM, 27-Aug-2012)

Ref Expression
Assertion in13 ( 𝐴 ∩ ( 𝐵𝐶 ) ) = ( 𝐶 ∩ ( 𝐵𝐴 ) )

Proof

Step Hyp Ref Expression
1 in32 ( ( 𝐵𝐶 ) ∩ 𝐴 ) = ( ( 𝐵𝐴 ) ∩ 𝐶 )
2 incom ( 𝐴 ∩ ( 𝐵𝐶 ) ) = ( ( 𝐵𝐶 ) ∩ 𝐴 )
3 incom ( 𝐶 ∩ ( 𝐵𝐴 ) ) = ( ( 𝐵𝐴 ) ∩ 𝐶 )
4 1 2 3 3eqtr4i ( 𝐴 ∩ ( 𝐵𝐶 ) ) = ( 𝐶 ∩ ( 𝐵𝐴 ) )