It's quite easy to show that two functions which satisfy the Cauchy-Riemann equations must be harmonic, but is the reverse true?

Specifically, if I have two functions u and v, which satisfy u_xx + u_yy=0 and v_xx+ v_yy =0, is it possible to show that they must also satisfy

u_x=v_y and u_y=-v_x??

Thanks for your help!!