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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

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    Mark13426, before the comments of others you may not have known that how you prove this depends on what formal proof system you use. Please don't feel discouraged that you didn't realize this. We all didn't know this at some point. That said, the very nature of a formal proof system requires that proofs in the object language stick rigorously to the rules and/or axioms of the system, or that slightly informal proofs can very easily get made into formal proofs (conversion of abbreviated wffs to actual wffs). Otherwise, the purported formal proof system is simply not formal.2012-04-12

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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.