In 5th ed would a bard who multi-class as a life domain cleric have his bard healing spells improved? [duplicate]

I have a 9th level bard. Hypothetically, if when I reached 10th level I took a level in cleric and chose the life domain, so I would be bard 9/cleric 1.

The multi-classing rules state "When you gain a new level in a class, you get its features for that level" and then lists a bunch of exceptions, none of which apply to this situation.

In the cleric class, at level 1 you take a domain. The life domain has an ability called disciple of life which reads

Also starting at first level, your healing spells are more effective. Whenever you use a spell of 1st level or higher to restore hit points to a creature, the creature regains additional hit points equal to 2 + the spell’s level."

Note that it says healing spells, not specifically cleric ones.

It would seem the RAW here is that this would apply to my bard spells as well. So my mass cure wounds (which is 5th level and I would not be able to know as a cleric spell but which I do know as a bard spell), would heal everything an additional 7 points.

Is this how multiclassing works? It seems like my best multi-class option if it does (although I’d probably wait until 11th level since level 10 bards get awesome stuff). Am I missing something?