$$\begin{array} {rc} \text{Assume:} & x > 2 \\ & \vdots \\ & \text{Some logic stuff} \\ & \vdots \\ \text{Conclude:} & x > 1 \\ \end{array}$$
Two common ways for this to be interpreted are (1) $$(\forall x~.~x > 2) \to (\forall x~.~x > 1)$$ and (2) $$\forall x~.~(x > 2 \to x > 1)$$. Logics that intend the first way include PRA and Hilbert style FOL. Logics that intend the second way include Fitch style Natural Deduction.
• Maybe something like global vs local ? In your first example the free var $x$ is implicitly universally quantified "locally", i.e. in the context of the single assumption, while in case (2) above it is quantified "globally", i.e. in the context of the entire derivation. – Mauro ALLEGRANZA Nov 13 '18 at 12:32