"A direct proof is a proof in which the truth of the premises of a theorem are shown to directly imply the truth of the theorem's conclusion."

Here are the premises:

(P -> R) ^ (Q -> S) ^ (~P) ^ (P v Q)

and the conclusion:

(S v R) ^ (~P)

Now what I do not understand why we are using expressions that are implications are not equivalency?

Let me start the prrof.

(1) ~P premise

(2) P v Q premise

(3) From discjunctive simplification we got:

(P v Q) ^ (~P) -> Q

(4) (Q->S) premise

(5) From detachment i.e (Q->S) ^ Q -> S

(6) P -> R premise

(7) from (6) ~P v R

And I didn't come up with the conclusion? What is the problem with this direct proof?

Shouldn't I use equivalent expressions and not implications?

Please help