Theorem inindi

Description: Intersection distributes over itself. (Contributed by NM, 6-May-1994)

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

Proof

Step	Hyp	Ref	Expression
1		inidm	⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴
2	1	ineq1i	⊢ ( ( 𝐴 ∩ 𝐴 ) ∩ ( 𝐵 ∩ 𝐶 ) ) = ( 𝐴 ∩ ( 𝐵 ∩ 𝐶 ) )
3		in4	⊢ ( ( 𝐴 ∩ 𝐴 ) ∩ ( 𝐵 ∩ 𝐶 ) ) = ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐴 ∩ 𝐶 ) )
4	2 3	eqtr3i	⊢ ( 𝐴 ∩ ( 𝐵 ∩ 𝐶 ) ) = ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐴 ∩ 𝐶 ) )