Question in the solution of an exercise in Van Dalen's "Logic and Structure" This is exercise 5 in Van Dalen's "Logic and Structure", and the question is to show:
$\Gamma \vdash \phi \implies \Gamma \cup \Delta \vdash \phi$
$\Gamma \vdash \phi ;  \Delta, \phi \vdash \psi \implies \Gamma \cup\Delta\vdash\psi$
I have two questions:

*

*What is this asking...? Can someone help me understand exactly what's going on in the question here?


*There is a solution given by Dr. Kevin T. Kelly's Logic and Computation course, in which he writes the following here:
I have a few questions on how this is written. Firstly, what is the set $\text{der}$ exactly? This is not defined in Van Dalen. Secondly, what does he mean when he writes "clamp $\cal{D'}$ onto $\phi$ in $\cal{D}'$"? Is this like the standard clamp function in analysis?
Cheers
 A: The two results above express the basic property of the derivability relation $\vdash$.
The symbol $\Gamma \vdash \phi$ means that there is a derivation $\mathcal D$ (in the Natural Deduction proof system) of conclusion $\phi$ from the set $\Gamma$ of assumptions (or premises).
In the linked comment, the existence of such a derivation is symbolized with: $\mathcal D \in \text {der}$.
Having said that, the first result says that: having a derivation $\mathcal D$ of formula $\phi$ from the set of assumptions $\Gamma$, we can add further assumption (the new set of formulas $\Delta$) and the derivation still holds.
This property is called Monotonicity of entailment.

The second result expresses the Transitivity of entailment.
About "to clamp"...?
The idea is simply this one: starting from derivation $\mathcal D$ of $\phi$ from $\Gamma$, add to the premises the new set $\Delta$ and copy-paste the derivation $\mathcal D'$ of $\psi$ from $\Delta$ and $\phi$ (that we have already as conclusion of $\mathcal D$).
The result is a new derivation $\mathcal D''$ of $\psi$ with the new set $\Gamma \cup \Delta$ of premises.
This property of the $\vdash$ relation formalizes the usual mathematical practice  of proving some preliminary Lemma to be used in the proof of a main Theorem.
