Finding the MLE for difference of two normally distributed means

Question

I was given that a random sample size n was taken from the following populations, N($\mu_{1}$,2) and N($\mu_{2}$,5). I am to find the MLE of $\tilde{\theta}$ for $\theta= \mu_{1}-\mu_{2}$. I have shown previously that for each individual distribution, the MLE would be $\hat{\mu_{1}}=\bar{x}$ and $\hat{\mu_{2}}=\bar{y}$. I believe $L=(\mu_{1},\mu_{2})$ would be multiplying their two likelihood functions together, which would net you a constant times the sum of their exponents. My question from here is what I should do to find the MLE from the original question? Since I know that $\hat{\mu_{1}}=\bar{x}$ and $\hat{\mu_{2}}=\bar{y}$, can I just say that $\tilde{\theta}$ = $\bar{x}-\bar{y}$, or do I need to do something more with the combined likelihood function. Thanks.