User blog comment:Emlightened/Little Bigeddon/@comment-24061664-20170117003234/@comment-5529393-20170117082653

∃!a means "there exists a unique a". ∃!a(φ(a))∧φ(k) means "there exists a unique a such that φ(a) is true, and also φ(k) is true."

You can always cut down the number of variables in a formula by bounding some of the variables, or defining some variables using others. So for example if we have a 6 variable formula w(a,b,c,d,e,f), we can define say x(a) = ∃b,c (w(u(a,b,c), v(a,c), y(a,b), a, b, c), and we will have a unary formula. So you really can get all the "expressivity" of the language for unary formulas, subject to the obvious restrict that it has to be unary formula.