PLEASE SOMEONE MUST KNOW HOW TO DO THIS!!! Logic Henkin/Completeness

I am so lost. Please please please, any help is so appreciated!!!

Call a set Σ of sentences in a language L Henkin′ in L if, for each formula φ and each variable x, if (iii) below holds, then (iv) also holds.
(iii) ∃xφ ∈ Σ.
(iv) φ(x; c) ∈ Σ for some constant c of L.
Let Σ* be a set of sentences in a language L* having properties (2) and (3) described in the statement of Lemma 4.3. Show that Σ* is Henkin in L∗ if and only if it is Henkin' in L*.

Also:
Lemma 4.3. Let Σ be set of sentences of a language L consistent in L. Let L∗ be gotten from L by adding infinitely many new constants. There is a set Σ∗ of sentences of L∗ such that
(1) Σ⊆Σ*;
(2) Σ* is consistent in L* ;
(3) for every sentence σ of L*, either σ belongs to Σ* or ¬σ belongs to Σ*;
(4) Σ* is Henkin.

More info:
Henkin sets. A set Σ of sentences in a language L is Henkin in L if, for each formula φ of L and each variable x, if (i) below holds, then (ii) also holds.
(i) φ(x; c) ∈ Σ for all constants c of L. (ii) ∀xφ ∈ Σ.

Hints:
For the “only if” direction, assume that Σ∗ is Henkin in L∗ and assume that (iii) holds (for Σ∗). You need to show that (iv) holds. Assume that (iv) does not hold. Use property (3) of Σ∗ to deduce that (i) holds for the formula ¬φ in place of φ. Use the assumption that Σ∗ is Henkin to get that (ii) holds for the formula ¬φ. Use property (2) of Σ∗ to show that this contradicts the assumption that (iii) holds for φ.
For the “if” direction, assume that Σ∗ is Henkin′ in L∗ and assume that (i) holds (for Σ∗). You need to show that (ii) holds. Assume that (ii) does not hold. Use properties (2) and (3) of Σ∗ to deduce that (iii) holds for ¬φ in place of φ. Use the assumption that Σ∗ is Henkin′ to show that (iv) holds for ¬φ in place of φ. Use property (2) of Σ∗ to contradict the assumption that (i) holds for φ.