Ordered semantic hyper-linking (OSHL) is a first-order theorem prover that tries to take advantage of the speed of propositional theorem proving techniques. It instantiates first-order clauses to ground clauses, and applies propositional techniques to these ground clauses. OSHL-U extends OSHL with rules for unit clauses to speed up the instantiation strategy. OSHL-U obtains many of the same proofs as Otter does. This shows that many first-order theorems can be obtained without true unification, so techniques used to speed up propositional provers may be applicable to them. OSHL-U, in finding proofs, also generates and stores significantly fewer clauses than resolution prover Otter on many TPTP problems. On some TPTP groups, OSHL-U finds more proofs than Otter, despite a slower inference rate.