help with this derivation

Question

need help with this one derivation problem please and thank

[(Ǝx)Cx v Fa) > (∀y)Hy,(Ǝx)Cx] |- (∀y)Hy v (Ǝz)Kz

Answer 1

For this proof you need to utilize the conditional that you have in premise 1.

First you need to construct the antecedent of premise 1 via vI:

1 | (((Ex)Cx v Fa) > (Ay)Hy) Premise

2 |_ (Ex)Cx Premise

3 | ((Ex)Cx v Fa) 2 vI

You then need to use line 3 to utilize the conditional in line 1:

1 | (((Ex)Cx v Fa) > (Ay)Hy) Premise

2 |_ (Ex)Cx Premise

3 | ((Ex)Cx v Fa) 2 vI

4 | (Ay)Hy 1,3 >E

Finally, you need to use vI to construct your conclusion:

1 | (((Ex)Cx v Fa) > (Ay)Hy) Premise

2 |_ (Ex)Cx Premise

3 | ((Ex)Cx v Fa) 2 vI

4 | (Ay)Hy 1,3 >E

5 | ((Ay)Hy v (Ez)Kz) 4 vI

Hope this helps!