$\forall \vec{b} \in \mathbb{R}^n, B \vec{x} = \vec{b}$ is consistent is equivalent to…

Suppose $A: n \times n$ and $B: n \times m$ and that $A$ is
invertible.

Prove that $\forall \vec{b} \in \mathbb{R}^n, B \vec{x} = \vec{b}$ is
consistent is equivalent to $\forall \vec{b} \in \mathbb{R}^n, (AB)
\vec{x} = \vec{b}$ is consistent.

I’m assuming this boils down to proving that $B$ and $AB$ are equivalent expressions, but I can’t see why this would be the case. Why does $A$ being invertible make the two statements equivalent (that is, why is $A$ invertible required)?

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