Theorem inabs3

Description: Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Assertion	inabs3	⊢ ( 𝐶 ⊆ 𝐵 → ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 ) = ( 𝐴 ∩ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		inass	⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 ) = ( 𝐴 ∩ ( 𝐵 ∩ 𝐶 ) )
2		sseqin2	⊢ ( 𝐶 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐶 ) = 𝐶 )
3	2	biimpi	⊢ ( 𝐶 ⊆ 𝐵 → ( 𝐵 ∩ 𝐶 ) = 𝐶 )
4	3	ineq2d	⊢ ( 𝐶 ⊆ 𝐵 → ( 𝐴 ∩ ( 𝐵 ∩ 𝐶 ) ) = ( 𝐴 ∩ 𝐶 ) )
5	1 4	eqtrid	⊢ ( 𝐶 ⊆ 𝐵 → ( ( 𝐴 ∩ 𝐵 ) ∩ 𝐶 ) = ( 𝐴 ∩ 𝐶 ) )