In functional analysis, the closed graph theorem is a good example of this. If you have two Banach spaces $X,Y$ and you define some linear operator $A : X \to Y$ on all of $X$, you often need to verify that it is continuous. Naively you would say, "let $x_n$ be a sequence in $X$ converging to some $x$; I have to show (1) that $A x_n$ converges and (2) its limit is $Ax$." But the closed graph theorem asserts that any everywhere defined operator between Banach spaces with a closed graph is in fact continuous. An operator has a closed graph iff for any sequence $x_n$ in $X$ such that $x_n$ and $A x_n$ converge, one has $\lim A x_n = A \lim x_n$. So in the argument above, you can skip the verification of (1) and treat it instead as an assumption.
There is also the closely related open mapping theorem, which says that a continuous linear bijection $B : X \to Y$ of Banach spaces is automatically a homeomorphism; i.e. you get to skip the verification that $B^{-1}$ is continuous. It's reminiscent of the theorem that any continuous bijection $f : K_1 \to K_2$ of compact Hausdorff spaces is automatically a homeomorphism.
