Given an implication $Pimplies Q$ its contrapositive is $neg Qimpliesneg P$ and its negation is $neg Pvee Q$.

What is the negation to $Pimplies Q$ when it comes to Linear Programming?

Can it be given an implication statement?
In my case I have $A(x,r)’leq bimplies r<0$ where $A$ is a rational matrix and $rinmathbb R$ and $x$ is a vector and $b$ is a vector.
The negation is $rgeq0wedge A(x,r)’leq b$.
Motivation:
I have a quantified Linear Program:
$forall xinmathbb R^m$
$exists rinmathbb R$
$A(x,r)’leq bimplies r<0$.
I want to write a converse statement which should be of form (to be in proper PRENEX form):
$exists xinmathbb R^m$
$forall rinmathbb R$
${rgeq0wedge A(x,r)’leq b}vee{“Something”}$.
What is the “something” to make the statement in PRENEX form?
 Can we give an implication statement to the converse by removing the $vee$?
Is the “something” $r<0$?
It does not make sense since the implication statement becomes
$$neg{r<0}implies{rgeq0wedge A(x,r)’leq b}$$
since clearly $neg{r<0}equiv{rgeq0}$ has nothing to do with $A(x,r)’leq b$.