Expectation Of a Poisson RV conditioned on several r.v.'s

Question

I have this problem: Given five independent Poisson Random variable (not necessary identical): $X_1$, $X_2$, $X_3$,$X_4$, $X_5$, need to find the conditional expectation: $\mathbb{E} \{ X_1 \mid X_2, X_1+X_2+X_3+X_4, X_1+X_2+X_3+X_5 \}$.

Here is what I have done:

1. Subtracting $X_2$ from the other given r.v.

   $= \mathbb{E} \{ X_1 \mid X_1+X_3+X_4, X_1+X_3+X_5 \}$

2. define $\alpha = X_1+X_3+X_4 $, $\beta= X_1+X_3+X_5 $

   $= \mathbb{E} \{ X_1 \mid\alpha, \beta \} = \mathbb{E} \{ X_1 \mid\alpha-\beta, \beta \}$ ( since we can always get back $\alpha$ and $\beta$ )

3. now noticing that $X_1$ is independent of $\alpha - \beta$ , we have :

   $= \mathbb{E} \{ X_1 \mid \beta \}$

   $= \mathbb{E} \{ X_1 \mid X_1+X_3+X_5 \}$ $= \mathbb{E} \{ X_1 \mid X_1+X_{3,5} \}$

Where I defined $X_{3,5} = X_{3}+X_{5}$, since $X_{3,5}$ is independent of $X_1$ we can use an earlier result that shows that the above is mean of Binomial distribution. Now my questions:

1. Is what I did above (especially step 1 and 2) correct?

2. Is there a more rigorous way to write step #2

3. If it is incorrect, can we get a close form for this expectation?