It's not altogether clear which axioms you are using, as there are different axiomatic systems. I will assume you can use DeMorgan's.
Note, as it seems that the left-hand side is universally quantified ($\forall$), and the right-hand side is the denial of an existential statement ($\lnot \exists$), you need to keep in mind that $\forall a:X(\text{blah})\iff \lnot \exists a:X(\lnot\text{blah}).\tag{$*$}$
$\text{Premise:}\quad \forall x:X\; \lnot (P \land Q)\tag{p}.$ $\forall x:X \;\lnot (P \land Q) \iff \forall x:X\;(\lnot P \lor \lnot Q)\tag{1.1}$ $\iff \lnot \lnot \left(\forall x:X \;(\lnot P \lor \lnot Q)\right)\tag{1.2}$ $\iff \lnot \exists x:X \;\lnot( \lnot P \lor \lnot Q))\tag{1.3}$
Step $(\text{p})\to (1.1)$ makes use of the equivalence $\lnot (P \land Q) \equiv (\lnot P \lor \lnot Q)$, by DeMorgan's;
Step $(1.1)\to (1.2)$ assumes $\lnot \lnot A \equiv A$ for any statement $A$;
Step $(1.2)\to (1.3)$ makes use of what I discuss at the start of this answer (see $(*)$).