D(1,3,4,5) !=0 and D(1,2,4,5) != 9.
D5 cannot be 0 (it would be lowest),
D5 cannot be higher than 2 because d5(3) + d4(6) => d1(9) but d3 is hi
+ghest.
d5 must be 1|2, so d4 must be 2|4, so d1 must be > 3, so d3 must be > 
+4. D2 must be 0|1.

Eliminate the impossible

Digit   1 2 3 4 5
-----------------
0       x 0 x x x 
1       x 1 x x 1
2       x x x 2 2
3       x x x x x
4       4 x x 4 x
5       5 x 5 x x
6       6 x 6 x x
7       7 x 7 x x
8       8 x 8 x x
9       x x 9 x x 

Leaves us these choices.

        4 0 5 2 1
        5 1 6 4 2
        6   7
        7   8
        8   9

d1+1 = d3 means this further reduces to

40521 
50621 
60721
70821
80921 *
70842 
71842 
80942 *
81942 *
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Eliminating those * that lack two prime digits leaves five⁺ six answers? (I think!)

(40521, 50621, 60721, 70821, 70842⁺, 71842)

⁺ Corrected. Thanks to crenz.