I was thinking about this question for a while and after some time, I did some research on it and found some sources.
For example, in $1.3.2$ of this(1) one, Myhill states that Power Set Axiom (which is impredicative) is stronger than its predicative counterpart Exponentiation Axiom and its result subset collection. There is also Separation Axiom in $1.3.1$ but both of them have a predicative counterpart although they are not stronger than the impredicative ones.
With a further research, I found this(2) one and got excited because of the second part "The identity of indiscernibles, Ramsey, and the counterexamples to the axiom of reducibility". Since axiom of reducibility roughly says that any propositional function can be expressed by a formally equivalent predicative truth function, I thought I could try to find a counter-example to that argument. However, I am not used to type theory and $(2)$ needs some to be understood. Therefore, I could not grasp the idea given there.
In short, can you give an impredicative definition, axiom or maybe a proof, preferably with a source, which hasn't got even weaker predicative counterpart? Thank you in advance.