The Collatz conjecture is equivalent to the following "induction principle":

If $P(0) \land P(1) \land (\forall{x} P(3 \cdot x + 2) \implies P(2 \cdot x + 1)) \land (\forall x P(x) \implies P(2 \cdot x))$,

then $\forall x P(x)$.

I am wondering if there are any statements that can be proved using this principle that are not obvious and are not obviously equivalent to the Collatz conjecture itself? I'm not so much interested in open problems (implications of the Collatz conjecture), rather something that may be easily provable a different way but also has a simple proof using "Collatz induction".

I have tried some statements $P(x)$ like "there exists a number of the form $2^a \cdot 3^b$ within a distance $f(x)$ from $x$" but I can't quite make this work.