Are the following definitions of a local property equivalent?

Question

I have seen two different definitions of a local property of rings.

1. $P$ is a local property of rings if $P(A)$ is equivalent to $P(A_{\mathfrak p})$ for all prime ideals $\mathfrak p$.

2. $P$ is a local property if $P(A)$ implies $P(A_f)$ for all $f\in A$, and $P(f_i)$ for $f_i$ such that $A=(f_1,f_2,...f_n)$ implies $P(A)$.

Here $P(A)$ denotes '$P$ holds for $A$'. Are these two definitions equivalent?