Limit and minimization

Question

For a given coefficient $\alpha \in \mathbb{R}$, I have to find a minimizer of $$ \min_{x \in \mathbb{R}, y \in \mathbb{R}} f(x,y) + \alpha [x \not= y], $$ where $[.]$ is the Iverson bracket, returning $1$ if its argument is true and $0$ otherwise. No assumption is made on $f$. My question is: how to mathematically demonstrate that when $\alpha\rightarrow\infty$, the above problem reduces to $$ \min_{x \in \mathbb{R}, y \in \mathbb{R}} f(x,y) $$ ? Then, minimizing over $x$ is enough since $x$ must be equal to $y$. $x$ and $y$ are in reality functionals but I do not know if it really matters. Thanks for your help.