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A simple example is obtained by taking $P$ to mean "has positive dimension".
Every local domain of positive dimension $(A,\mathfrak m)$ has $P$ at all maximal ideals (i.e. just at $ \mathfrak m$ !) since $A_{\mathfrak m}=A$ , but $P$ fails at the generic point $\eta=(0)$ since $A_\eta=Frac(A)$ has dimension zero, being a field.