I did
exactly the same when I was in therapy the time before last. The only person who knew where I actually was, was my Manager (and I guess HR as well) - but my team (peers and subordinates) didn't know. I felt at the time this was the correct way to handle it.
Third time around in therapy (last year), I decided to be very open about my mental health, and the steps I was taking to help myself. So my emails became 'I can't do that then, I'm in therapy.'.
I found personally, that being honest and open to everyone about my problems actually contributed in a small way to my recovery effort. Even now, stabilised and medicated, I'm still honest and open about my [ongoing] journey, and who I was as person then, and who I am now.
I am never going to lie about being crazy again, it's not healthy.