First-order tableaux

Prove each of the following (where $\phi$ $\dashv$ $\vdash$ $\psi$ iff $\phi$ $\vdash$ $\psi$ and $\psi$ $\vdash$ $\phi$):

1. $\exists$x[Fx $\wedge$ $\neg$Fx] $\vdash$
2. [$\exists$xFx $\wedge$ $\forall$x$\neg$Fx] $\vdash$
3. $\exists$x$\forall$y[Rxy $\leftrightarrow$ $\neg$Ryy] $\vdash$
4. $\forall$xFx $\dashv$ $\vdash$ $\neg$$\exists$x$\neg$Fx
5. $\exists$xFx $\dashv$ $\vdash$ $\neg$$\forall$x$\neg$Fx
6. $\forall$x[$\exists$yFxy $\rightarrow$ Gx] $\dashv$ $\vdash$ $\forall$x$\forall$y[Fxy $\rightarrow$ Gx]
7. $\forall$x$\exists$y[Fax $\wedge$ Gay] $\dashv$ $\vdash$ $\exists$y$\forall$x[Fax $\wedge$ Gay]