Number conversion via Prime route

Question

This is a variation of the Word Ladder. Instead this is a number ladder.

Convert the number 12345 to the number 54321 in seven or less steps with the following rules

A: You can only change any one digit at a time using digits 1 to 9

B: After the change the new number formed must be a Prime number

C: The Prime number cannot have a repeat digit in it. All five digits must be different.

D: You can check the Prime Number list from the computer but no programming please.

Bonus Question

Will you shorten the conversion if Rule C is waived?

Answer 1

Here is a seven step solution which is the minimum

12345 -> 12347 -> 92347 -> 98347 -> 98327 -> 94327 -> 94321 -> 54321

Proof that this is minimal

Jaap Scherphuis makes the point in the comments that it would take at least three steps to swap the 2 and 4 and at least three steps to swap the 1 and 5, under the rules, so let's see if it can be done.
You have to change the last number first because numbers ending in 5 won't be prime and the only choice here is 7, so we get 12347. To achieve six steps, we then must be able to swap the 2 or 4 with a digit not already used or the 1 to a 5 but all cases result in a composite number, so six steps cannot be achieved.

Further to this,

With the idea that we must change 1, 2 or 4 next, the only possibility is that the 1 must change to a 6 or a 9. To achieve a seven step solution, the next step must be changing the 2 or 4 to something already unused or changing the 7 to a 1. The 62347 proves fruitless in this regard but the 92347 path is very fruitful and it's not too hard to generate a solution on this path (I found a few).

What if Rule C is waived

12345 -> 12347 -> 14347 -> 14327 -> 14321 -> 54321.
This is five steps and this is optimal because changing the 5 to 7 is a necessary first step and from there, since four digits are still different, we cannot do better than four steps.

Note: Jaap Scherphuis makes the very valid point in the comments that we can also change the 5 to a 3 in the first step (I missed this) but that it would also take four more steps to convert to our final answer (for the same reasons as above).
Here is a possibility in this case
12345 -> 12343 -> 12323 -> 14323 -> 54323 -> 54321