Theorem inindir

Description: Intersection distributes over itself. (Contributed by NM, 17-Aug-2004)

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

Proof

Step	Hyp	Ref	Expression
1		inidm	⊢ ( 𝐶 ∩ 𝐶 ) = 𝐶
2	1	ineq2i	⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐶 ∩ 𝐶 ) ) = ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 )
3		in4	⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐶 ∩ 𝐶 ) ) = ( ( 𝐴 ∩ 𝐶 ) ∩ ( 𝐵 ∩ 𝐶 ) )
4	2 3	eqtr3i	⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 ) = ( ( 𝐴 ∩ 𝐶 ) ∩ ( 𝐵 ∩ 𝐶 ) )