Deduction rules involving set $\Gamma$ of premises versus elementary textbook natural deduction rules. How do they differ exactly? In elementary textbooks, natural deduction rules are presented in the following way, say, for $\&$-Intro
from $\phi$ and $\psi$, infer $\phi\&\psi$
or
$(n).....\phi$
$(m)....\psi$
$\therefore$
$(p)....\phi\&\psi$.
I would like to know to what extent does the following way of stating $\&$-Intro differs from the above  " ordinary" textbook presentation. The way I'm referring to is the one I find in Shapiro's presentation of classical logic (https://plato.stanford.edu/entries/logic-classical/#Dedu):
(&I) If Γ1⊢θ and Γ2⊢ψ, then Γ1,Γ2⊢(θ&ψ).
( meaning : " if $\theta$ is derivable from a set of premises $\Gamma_1$ anf if $\psi$ is derivable from a set of premises $\Gamma_2$, then $(\theta\&\psi)$ is derivable from a set of premises $\Gamma_1\cup\Gamma_2$.")
Can Shapiro's presentation be termed " natural deduction"? Or rather is it a case of " sequent calculus"?
Aside : Do you know any beginner's textbook on mathematical logic that displays examples of derivations in Shapiro's style?
 A: The 'elementary textbook' rule is that: when $\phi$ and $\psi$ may be derived, then we may infer that $\phi\mathop\&\psi$ may be derived.   It is unstated that these derivations take place in the same context ( premises and assumptions ).   This rule of inference might be sumarised as $$\dfrac{~\phi\qquad\psi~}{\phi\mathop\&\psi}{\small\&\mathsf I}$$
The 'sequent calculus' rules extend this to explicitly list under which context things are derived.   The same rule above may be then presented with the context ($\Gamma$, a set of statements) explicitly stated: $$\dfrac{~\Gamma\vdash\phi\qquad\Gamma\vdash\psi~}{\Gamma\vdash\phi\mathop\&\psi}{\small\&\mathsf I}$$
We can then extend the rule so to say: when $\phi$ and $\psi$ may be derived in contexts $\Gamma_1$ and $\Gamma_2$ respectively, then we may infer that $\phi\&\psi$ may be derived in the united context, $\Gamma_1\cup\Gamma_2$.
$$\dfrac{~\Gamma_1\vdash\phi\qquad\Gamma_2\vdash\psi~}{\Gamma_1\cup\Gamma_2\vdash\phi\mathop\&\psi}{\small\&\mathsf I}$$

Long story short: the advanced presentation says the same thing as the elementary presentation, but with some additional details added.
