Natural Deduction - 'monotone' property of sequent

Question

In natural deduction, what says that the following is correct?

$\Gamma \Rightarrow B$ then $ \Gamma, A \Rightarrow B$

I saw a proof that uses this rule without mentioning it and I can't find the rule/axiom/proof for it.

Answer 1

See Chiswell & Hodges : Natural Deduction has the Monotonicity property :

If $Γ ⊆ Δ$ and $Γ \vdash \varphi$, then $Δ \vdash \varphi$.

Suppose $Γ \vdash \varphi$. This means that there is a derivation $D$ whose conclusion is $\varphi$ and whose undischarged assumptions are all in $Γ$.

Given that $Γ ⊆ Δ$, the undischarged assumptions of $D$ are also in $Δ$, so that $D$ proves $Δ \vdash \varphi$.