Prime challenge

As the given prime numbers are all different, the first condition implies that $A, B, D, E$ are all odd. The third condition then implies that $C$ is even, i.e. $C=2$ .

Using $C=2$ and $A=B+2$ we can eliminate $A$ and $C$ and are left with the following three conditions:

$2 \times B + D + 2 = E$

$2 \times B + 7 =E$

$3 \times B - 10 < E$

The first two conditions imply $D = 5$ . The second two conditions imply $B < 17$ . Taking into account that $A= B + 2$ also needs to be a prime number and $A$ and $B$ need to be different to $D=5$ we can conclude $B=11$ and hence $A=13$ and $E=29$ .