[DanBC]

To support what you say here's the nice guidance for problem drinking combined with mental health problems. The guidance is very clear: you treat an alcohol addiction before you treat the depression because the depression is likely to become less severe as you treat the problem drinking.

http://www.nice.org.uk/guidance/cg115/chapter/Key-priorities...

> For people who misuse alcohol and have comorbid depression or anxiety disorders, treat the alcohol misuse first as this may lead to significant improvement in the depression and anxiety. If depression or anxiety continues after 3 to 4 weeks of abstinence from alcohol, undertake an assessment of the depression or anxiety and consider referral and treatment in line with the relevant NICE guideline for the particular disorder[3].

So, it's true for the general population. But it's a bit rude to ask one individual if they had problem drinking or to suggest their illness was problem drinking.