Page "Original proof of Gödel's completeness theorem" Paragraph 18

Next, we eliminate all free variables from φ by quantifying them existentially: if, say, x < sub > 1 </ sub >... x < sub > n </ sub > are free in φ, we form.

If ψ is satisfiable in a structure M, then certainly so is φ and if ψ is refutable, then is provable, and then so is ¬ φ, thus φ is refutable.

We see that we can restrict φ to be a sentence, that is, a formula with no free variables.