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The first is "uniqueness is a relative form of existence", due apparently to Shmuel Weinberger. This slogan seems to appear frequently in operator theory. Take, for example the problem of proving that K-theory commutes with direct limits (say, of C* algebras $A_1 \subseteq A_2 \subseteq \ldots \subseteq A$). There are two components to the proof: surjectivity (the "existence" part) which amounts to showing that every element of $K_0(A)$ lies in the image of some $K_0(A_j) \to K_0(A)$, and injectivity (the "uniqueness" part) which involves proving that if two elements of $K_0(A_j)$ are equivalent in $K_0(A)$ then they are equivalent in $K_0(A_j)$. Once you have proven existence you can verify uniqueness by joining representatives of your chosen $K_0(A_j)$ classes by a homotopy in the space of generators for $K_0(A)$ and then use your existence argument to lift to a homotopy in $A_j$. In other words, prove uniqueness by applying your existence argument to a pair.
The second is the (in)famous "Eilenberg Swindle" which seems to come up everywhere. I first encountered it in K-theory, but I think the canonical example is the argument which proves that the $n$-sphere is prime with respect to connected sum (which I will denote +). Suppose that $M$ and $N$ are manifolds such that $M + N = S^n$. We have that $(M + N) + (M + N) + (M + N) + \ldots$ is homeomorphic to $\mathbb{R}^n$ (it is a cylinder with the left opening glued shut), and similarly so is $(N + M) + (N + M) + \ldots$. Since $M + (N + M) + \ldots = (M + N) + (M + N) + \ldots$, we have shown that $M + \mathbb{R}^n = \mathbb{R}^n$ which forces $M$ to be homeomorphic to $S^n$.