Quick description
To prove a property for all , first show that is equivalent to for all in the desired parameter space. Then, one only needs to verify for a single , which one can choose to make the verification as easy as possible.
Prerequisites
Probability theory
Example 1
(Lindeberg replacement trick) Suppose one wants to prove the central limit theorem, viz. that if is a sequence of iid realvalued random variables with mean zero and variance , then the random variables converge in distribution to the standard Gaussian random variable . For simplicity let us assume that all moments of the are finite. Lindeberg's proof of the central limit theorem proceeds in two steps:

(Base case) Verify the central limit theorem in the special case when the are iid gaussians, . In this case the theorem is easy, basically because the sum of two independent gaussians is still a gaussian.

(Invariance) Show that for each , the asymptotic limit of the moments of remain unchanged if one replaces the iid sequence by any other iid sequence, say , with the same mean and variance. This is done by expanding out and observing that most terms only involve at most two factors of each , and so (by the iid hypothesis) only involve the first and second moments of the .
Indeed, once one has the invariance principle, one simply replaces the with a gaussian iid sequence and uses the base case.
General discussion
The replacement trick has also been used in random matrix theory; see this blog post for some further discussion.