Proof Mind is Fundamental and Matter Doesn't Exist

[Rational AKD]

Your premise 3 contains a fallacy.

Premise 3: "The mind" has property x, while "matter" does not have property x.
Property x is: "If the world is solipsistic, this thing can still exist."
There are two options:
1: The world is solipsistic. Then, "the mind" exists and "matter" does not.
2: The world is not solipsistic.  Then, because the property you proposed contains a falsity as its premise, it conveys no information.  Thus, property x is not a real distinction between "the mind" and "matter."
That is to say, if the world is solipsistic, then clearly the mind and matter are different; if the world is not solipsistic, then the difference between "the mind" and "matter" - which is subtly premised on the assumption of a solipsistic world - no longer exists.

I believe this refutes the argument.

premise 3 is derived from Leibniz Law of Indiscernibility of Identicals. given A and B; A is identical to B if and only if for all things true of A, B. so if anything is true of A but not of B, then A and B are not identical. so, given it is possible for mind to exist in a solipsist (immaterial) world, but it is not possible for matter... then there is something true of mind that is not true of matter. and yes, Leibniz Law covers modal truths as well.