Logical content of Gauss's Lemma (arithmetic)

In the context where $a$, $b$, $c$ are integers we have $(a \mid bc, a\land b = 1) \Rightarrow a\mid c$. This result is called Gauss's Lemma in French Highschool.It is well known that (Steve Awodey, Category Theory) the divisibility relation $a \mid b$ can be interpreted as the logical implication $a \to b$. In this setting, the gcd $a \land b$ becomes the associated conjunction $a$ AND $b$ with the universal property : $(c \to a, c \to b) \vdash c \to a \land b$. Dually, the lcm $a \lor b$ is interpreted as the disjunction.

My question : in this setting, how would you read Gauss's Lemma given above ?

Bricks for the answer :

1. $\forall a, 1 \mid a$ reminds me of the ex falso quod libet (explosion principle) also reading $\bot \to P$ (from false deduce anything) ;
2. $a \land b = 1$ would thus be read NOT($a$ AND $b$) ?
3. if $d = b \cdot c$, $b$ becomes a proof that $c \mid d$.

From there on, I'm not yet able to conclude anything worthy, but I'm sure there's something….