Theorem ssinss2d

Description: Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypothesis	ssinss2d.1	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
Assertion	ssinss2d	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ssinss2d.1	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
2		incom	⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 )
3	1	ssinss1d	⊢ ( 𝜑 → ( 𝐵 ∩ 𝐴 ) ⊆ 𝐶 )
4	2 3	eqsstrid	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )