Theorem in13

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

	Ref	Expression
Assertion	in13	⊢ ( 𝐴 ∩ ( 𝐵 ∩ 𝐶 ) ) = ( 𝐶 ∩ ( 𝐵 ∩ 𝐴 ) )

Proof

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