Proof

By hypothesis, $m\mid (a-b)c$ for some integers $a,b,c$ with $m > 1$ being a positive integer.
From the divisibility laws (law 7) it follows $$\frac{m}d\mid (a-b)\frac{c}d,$$ where $d=\gcd(c,m)$ is the greatest common divisor of $c$ and $m.$
From generating co-prime numbers knowing the $\gcd$, it follows that $\frac{m}d$ and $\frac{c}d$ are co-prime.
By divisors of a product of two factors, co-prime to one factor, it follows $$\frac{m}d\mid (a-b).$$
This is equivalent to $a\left(\frac{m}d\right)\equiv b\left(\frac{m}d\right).$
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Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927