Is this application of the law of total probability correct?

Question

Let us consider a counting process $N(t)_{t\geq0}$ which is neither Markovian nor Levy. Is it correct to write

$$ \mathbb{P}(N(t)=j)=\int_{0}^{t}\mathbb{P}(N(t)=j, N(s)=i)ds $$

for $j\geq 1$ and $i\leq j$?

Thanks in advance.