I want to test a set of four hypotheses. I estimate a single model to test all of them (each hypothesis pertains to one coefficient in the model). I repeat this estimation 10 times with 10 slightly different definitions of the dependent variable. I want to reject each null hypothesis if at least one of the 10 repeated tests indicates significance for the respective coefficient.

How do I correctly adjust the p-values in this case? Do I have to adjust them twice? My two concerns are the following: First, do I have to adjust them based on the fact that I test four hypothesis with one model? If this is the case, wouldn´t I always have to adjust any p-value in a multiple regression model then? Because in a multiple regression model, one always implicitly tests wether multiple coefficients are different than zero, i.e. tests multiple hypotheses. And second, do I have to adjust them to account for the fact that I repeat this test 10 times? 

My assumption would be that I would have to adjust them once to account for the 10 repeated tests, is this correct?