First-order validity

Either prove or provide a counterexample to each of the following:

1. $\exists$x[Fx $\wedge$ Gx] $\vdash$ [$\exists$xFx $\wedge$ $\exists$xGx]
2. $\exists$x$\exists$yFxy $\vdash$ $\exists$xFxx
3. $\exists$xFxx $\vdash$ $\exists$x$\exists$yFxy
4. [$\exists$xFx $\leftrightarrow$ $\exists$xGx] $\vdash$ $\forall$x[Fx $\leftrightarrow$ Gx]

Readings

(Hodges, 2001, §§39-41)