i am trying to solve a form of the navier stokes equation. The velocity is a 2 dimensional stream flow ie v=(∇×Ψ)j=v=(-Ψ_{z},0,Ψ_{x})

i want to simpify the following expression using the fact that the divegrence of the velocity is zero but i dont know what identity to use,
the goal is to get an expression for ∇⁴Ψ
i have ∇×∇²v=∇×∇²(∇×Ψ)

i know that ∇×(∇×Ψ)=∇(∇⋅Ψ)-∇²Ψ
and this =-∇²Ψ in this case since the divergence is zero but how does the laplacian in fron change the identity?

can i simply make it ∇×∇²(∇×Ψ) = ∇²(∇⋅Ψ)-∇⁴Ψ
is this a valid identity?