nt.number theory - Strengthening of Dirichlet's theorem on arithmetic progressions

Hello all, Dirichlet's famous theorem asserts that any arithmetic progression
$lbrace ax+b | x in {mathbb N}rbrace$ contains infinitely many primes if a
and b are relatively prime.

I am wondering if the following strengthening of Dirichlet's theorem is also
true :

Let $a,b$ be relatively prime integers
as above. Then there is a uniform bound $g(a,b)$ such
that any interval $lbrace x+1,x+2, ldots ,x+g(a,b)rbrace$ of $g(a,b)$
successive integers contains at least one integer $y$ which is
congruent to $b$ modulo $a$ and which is not divisible by
any integer between $x+1$ (inclusive if $yneq x+1$) and
$y$ (exclusive).

Without the uniform bound, this would be a tasteless easy consequence
of Dirichlet's theorem. With the bound, however, it becomes stronger
than Dirichlet's theorem.

Perhaps the two are in fact equivalent ?