A general strategy

The Principle Lemma implies the Completeness Theorem. To show this, suppose that the Principle Lemma is true and consider a set Γ and a sentence P such that Γ |= P. We need to show that Γ ` P. Well, since Γ |= P it follows that Γ∪{∼P} is not truth-functionally consistent (you proved this in a homework question). But then by the Principle Lemma it follows that Γ∪{∼P} is not consistent in SD (if you don’t see this at first, look at the alternative expression of the Principle Lemma described in footnote 1). But if Γ∪{∼P} is not consistent in SD then Γ ` P (you proved this in a homework question). So Γ ` P, as required.