Let $x,y$ be variables, $A(x,y)$ a formula in which both $x$ and $y$ occur free.
Show that
$\forall x \Big(\forall y\big(A(x,y)\big)\Big) \to \forall y \Big(\forall x\big(A(x,y)\big)\Big)$
is logically valid
Let $x,y$ be variables, $A(x,y)$ a formula in which both $x$ and $y$ occur free.
Show that
$\forall x \Big(\forall y\big(A(x,y)\big)\Big) \to \forall y \Big(\forall x\big(A(x,y)\big)\Big)$
is logically valid
In any structure $M$, both formulas hold if and only if $M \models A(x,y)$ for all $x,y \in |M|$. Thus $M \models (\forall x)(\forall y)[A(x,y)] \leftrightarrow (\forall y)(\forall x)[A(x,y)]$. Because that last formula is satisfied by every structure, by definition it is logically valid.