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All the proofs of the work-KE theorem I have found go something like this:

W= int(F)dx from x1 to x2

= m(int(dv/dt))dx from x1 to x2

= m(int((dv/dt)v)dt from t1 to t2

= (m/2)(vf^2-vi^2) by the chain rule/substitution

My question is regarding the substitution "dx=vdt," and the associated change of variables from x to t, which is obviously true but how do you justify it from the substitution rule or something else not involving differentials? (I'm sure that expression makes perfect sense when it comes to differential forms but I have not learned those yet)