dnd 5e – Can a multiclass Artificer use Alchemical Savant on spell learned through the other class(es)?

The description for Alchemical Savant say:

You’ve developed masterful command of magical chemicals, enhancing the healing and damage you create through them. Whenever you cast a spell using your alchemist’s supplies as the spellcasting focus, you gain a bonus to one roll of the spell. That roll must restore hit points or be a damage roll that deals acid, fire, necrotic, or poison damage, and the bonus equals your Intelligence modifier (minimum of +1).

Now, say I have multiclassed with a Wizard and learn Infestation and Chromatic Orb via that class. Both can do poison damage so they qualify on that part.

Can I take advantage of Alchemical Savant with either spell?

Unlike a lot of other class features, it doesn’t say, “When you cast an (insert class name) spell…”, just cast a spell and use tools as the focus.

Initially I would just say no, but, while not laid out in the PHB, in XGE it says that the Alchemist’s Supplies kit contains:

Alchemist’s supplies include two glass beakers, a metal frame to hold a beaker in place over an open flame, a glass stirring rod, a small mortar and pestle, and a pouch of common alchemical ingredients, including salt, powdered iron, and purified water.

And in the PHB, the (Wizard’s Arcane) focuses are described as:

An arcane focus is a special item–an orb, a crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item–designed to channel the power of arcane spells. A sorcerer, warlock, or wizard can use such an item as a spellcasting focus.

So it may be possible to have an arcane focus built-in to the character’s alchemist’s supplies and do double duty…

The is a related question about spells with no material components, but this is about spells that have material components, but learned through a different class.