Count unique values in one column given multiple criteria for other columns

I looked through the forum, and couldn't find anything exactly like my problem.

I cannot use VBA.

My problem is this:

I have a spreadsheet with multiple columns: Column A, B, C, D, etc. I need to know the sum of UNIQUE values in column A where certain conditions are met in Columns C, D, E, etc.

For one row, I expect the forumula would look something like the following:

If (Column B == "crit1"){
if (Column C == "crit2"){
if (Column D != "crit3"){
count_unique_values(Column A);
}
}
}

Of course, nothing like this will work since my sheet has multiple rows with a mix of values.

Note that only UNIQUE values in column A shall be counted.

How could I expand the equation to include additional criteria in other columns?

Thanks in advance! Please let me know if there are any questions.