$$\forall y P(x) \implies \exists y Q(x,y)$$
Is it valid to rewrite the first part of the implication as $P(x)$ since the variable being quantified has nothing to do with the statement?
2026-03-29 15:53:06.1774799586
Quantifying over a variable that is not in the statement?
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Yes. If $y$ does not appear free in $P(x)$, then $\forall y P(x)$ and $P(x)$ are logically equivalent.