It's predicate logic and I need to find a counterexample to disprove the follwowing claim
$(A \models \phi \implies A \models \psi) \implies A \models \phi \rightarrow \psi$
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Some people define a formula (with free occurrences of variables) to be true in the structure $A$ if the sentence obtained by universally quantifying these variables is true in $A$. Then annoying things can happen, which is why I prefer not to do it.
For an informal example, let $A$ be the natural numbers, let $\phi(x)$ be the formula that says $x$ is even, and let $\psi(x)$ be the formula that says $x$ is odd. (It is not hard to write down the appropriate formulas.) Then it is not the case that $\phi$ is true in $A$. It follows that $A \models \phi \implies A \models \psi$ is true. But $ A \models \phi \rightarrow \psi$ is false, since it is not the case that for all $x$, if $x$ is even then $x$ is odd.
Remark: Expressing commutativity of addition as $x+y=y+x$ rather than $\forall x\forall y(x+y=y+x)$ is a useful abbreviation. However, building that abbreviation into the logic introduces complications, as illustrated by the example above.