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Is there a potential for loss of information to occur when we simplify an expression?

As an example, lets say I have the expression

$\forall y P(x)\tag{1}\label{1}$

If y does not appear free in P(x), I am aware that I may write $\forall y P(x)$ simply as $P(x)$.

However if another mathematician were to use my final result "$P(x)$", and then allowed $y$ to be free in $P(x)$, the simplified result will no longer follow the unsimplified one. It appears that the final result should contain some reference to what is not allowed.

i.e. $P(x)$ ,where $y$ is not free in $P(x)$. $\tag{2}\label{2}$

However, ($\ref{2}$) is more restrictive than ($\ref{1}$).

It seems to me that a person would be better off not simplifying the expression because it allows more freedom when working with the result.

My question is, should the final result contain some reference to what is not allowed? And if so; is there a potential for loss of information to occur when we simplify the expression, and then write such a reference?

http://chat.stackexchange.com/transcript/message/35041891#35041891

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    You have to consider the "formal specifications" of the syntax and the *abbreviations* we usually use in the meta-theory for brevity (and readbility). In the formal language, if $P$ is a *predicate* symbol wit arity $1$ (i.e. a unary predicate symbol) it **must** be written $P(x)$: there are no "omitetd" variables. A quantified expression $\forall yP(x)$ is different from $\forall yP(x)$: they are different strings. When we consider the *semantics* (the rules for interpreting formulae), we have that $\forall yP(x)$ has the "same meaning" as $P(x)$: but we have to prove it. – 2017-01-27
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    In the meta-theory, it is useful to use meta-variables: usually Greek letters for formulae: $\varphi$. The "convention" is that if we write $\varphi(x)$ we mean a formula with at least a free occurrence of $x$; this does not imp.ies that $x$ is the only free variable. Usually, we do this when specifying the inference rule, like e.g. $\forall x \varphi(x) \vdash \varphi(t)$. In this case the rule tell us how to eliminate the leading quantifier, irrespective of the "inner structure" of the formula. – 2017-01-27
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    what do you mean by "if P is a predicate symbol wit arity 1 (i.e. a unary predicate symbol)" are wit, arity and unary terminology or have you made a small spelling error? – 2017-01-27
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    $\text {Even}(x)$ is a unary predicate; $x < y$ is a binary predicate; "$y$ lies between $x$ and $z$" is a ternary predicate. [Arity](https://en.wikipedia.org/wiki/Arity) is the "general" name for the number of arguments of the predicate (or relation or function). – 2017-01-27
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    I should clarify that Y was not omitted in P(x); but it was added in by another mathematician when they tried to extend the theorem. If the final result of (1) included ∀y but did not depend on it at that time; the new mathematician would not need to do any work because they could add y in (knowing how the workings would turn out). If the expression was simplified this would not be the case. – 2017-01-27
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    I hope that makes sense – 2017-01-27
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    If you stay with "formal" symbolic expressions, $P(x)$ is $P(x)$ (unary) and you cannot change it to $P(x,y)$ (binary). – 2017-01-27

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