This question has essentially been answered in the comments, so I see no reason for it to remain officially unanswered.
The proposed property holds. It is "equivalent to" Dirichlet's theorem, in the informal sense that each could be used to prove the other. However, we are not talking about axiomatization, so rather I would say that both are true and equivalence isn't meaningful. (This is not to be confused with Fréchet's objection to Tarski.)
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