No you can't conclude this and in fact the more general principle is that being able to prove P(0), P(1), P(2), etc... does not mean that it's also possible to prove P(n) for all n. The property of a formal system for which this does follow is known as Omega completeness, but Peano arithmetic and ZFC more broadly are both Omega-incomplete systems, hence there are propositions for which P(0), P(1), P(2)... can all individually be proven and yet it is not possible to prove P(n) for all n.

Wikipedia touches on this subject with an article on omega-consistent systems, which is a very closely related property:

https://en.wikipedia.org/wiki/%CE%A9-consistent_theory

And this gives a very brief description of Omega completeness:

https://encyclopediaofmath.org/wiki/Omega-completeness