# Why not weaken reducibility in $K_2(W)$ instead of sub-world separation?

In this theory $$K_2(W)$$ (page 7) Harvey Friedman argue for weakening Sub-world Separation into SS-. He did that in order to evade making $$W$$ transitive. But he could have done that by simply reverting to the more natural way, which is the origin of the Reducibility axiom (which is a kind of reflection principle), that of changing Reducibility to the following:

if $$\varphi$$ is a predicate definable by a formula in $$L(\in)$$ from parameters $$x_1,..,x_n$$, then all closures of:

$$x_1,..,x_n \in W \wedge \varphi(W) \to \exists x_{n+1} \in W (\varphi(x_{n+1}))$$; are axioms.

I mean if we keep sub-world separation as it is in $$K(W)$$, (i.e.; don't weaken it to SS-), and replace Reducibility with the above, and keep the other axioms of $$K_2(W)$$ as they are, call the resulting system $$K^* _2(W)$$, would that be weaker than $$K_2(W)$$ or otherwise be inconsistent?