Theorem in31

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

	Ref	Expression
Assertion	in31	⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 ) = ( ( 𝐶 ∩ 𝐵 ) ∩ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		in12	⊢ ( 𝐶 ∩ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐴 ∩ ( 𝐶 ∩ 𝐵 ) )
2		incom	⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 ) = ( 𝐶 ∩ ( 𝐴 ∩ 𝐵 ) )
3		incom	⊢ ( ( 𝐶 ∩ 𝐵 ) ∩ 𝐴 ) = ( 𝐴 ∩ ( 𝐶 ∩ 𝐵 ) )
4	1 2 3	3eqtr4i	⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 ) = ( ( 𝐶 ∩ 𝐵 ) ∩ 𝐴 )