The question is whether the following argument holds:

I start with some model \mu_1(x,c) = E[y \lvert x,c]

such that E[y\lvert x] = \int_c E[y \lvert x,c] p(c\lvert x) dc

I then assume that c=h(x,v) such that for any given x I have a function c=k(v) (I am specifically thinking about c=vx).

Further more I assume that p(v\lvert x) = p(v). Can I then apply integration by substitution to get:

E[y\lvert x] = \int_c E[y \lvert x,c] p(c\lvert x) dc = \int_v E[y \lvert x,k(v)] p(v)  dv?

Hope this makes sense otherwise let me know. Any input much appreciated