You've misapplied two logical rules.
First, a conditional statement is not logically equivalent to its inverse.
Consider the claim, "It rained, so my grass is wet." Suppose it is true. The inverse is, "It did not rain, so my grass is not wet." This is not necessarily true, since the sprinkler could have gone off. You may be looking for the contrapositive.
Second, you are not correctly "distributing" the negation throughout the OR statement. That is,
~(A OR B) == ~A AND ~B.
To see why this is true, let A be "it is raining" and B be "it is summer". If we have (A OR B) we are claiming that it is either raining or it is summer (or both). If we have ~(A OR B) we are claiming that is not the case that it is raining or it is summer. That is, we are claiming that it is not raining AND it is not summer.
A → B is not equivalent to ~A → ~B. [inverse]
A → B is equivalent to ~B → ~A. [contrapositive]
~(A OR B) == (~A AND ~B) [De Morgan's Law]
To your point, the following would be a correct logical sequence:
P(x) → (∃xP(x) ∨ ¬∃xP(x))
¬(∃xP(x) ∨ ¬∃xP(x)) → ¬P(x)
(¬∃xP(x) ∧ ∃xP(x)) → ¬P(x)
The final statement is vacuously true, since its antecedent is apparently always false. (And any conditional with a false antecedent is defined to be true.)