Prove that: $\vdash \forall x(\forall y\alpha)\to \forall y(\forall x \alpha)$ How does one prove that

$$\vdash \forall x(\forall y\alpha)\to \forall y(\forall x \alpha)$$

in first order logic?
I have tried using the specialization and generalization rules on various wffs but they don't lead me to anything concrete.


I have tried using the specialization and generalization rules on various wffs but they don't lead me to anything concrete.

They should.
Just assume $\forall x~(\forall y~\alpha)$, instantiate using fresh arbitrary variables, and generalise in reverse order to derive $\forall y~(\forall x~\alpha)~$, under that assumption.
Now you may deduce $(\forall x~(\forall y~\alpha))\to(\forall y~(\forall x~\alpha))$.
$\blacksquare$
