Function of quantile function

Question

I stumbled across this Theorem in a paper, but I am not able to obtain its reference or solve it. Please help.
Let $X_1, ..., X_n$ be independent random variables with CDF $\Phi_1,...,\Phi_n$ (which are continuous and strictly increasing), then $X = f(X_1,...,X_n)$, where $f$ is strictly increasing, is a random variable with inverse CDF $$ \Phi^{-1}(\alpha) = f(\Phi_1^{-1}(\alpha),...,\Phi_n^{-1}(\alpha)). $$ I thought it would be simple if we use copula, but it works with joint and marginal CDF, here we have quantile function.

Answer 1

This is false. If $X_1, X_2$ are independent $N(0, 1)$, then their sum $X = X_1 + X_2 \sim N(0, 2)$, with quantile function $\Phi^{-1}(\alpha) = \sqrt{2}\Phi_1^{-1}(\alpha)$. However, the result you have states that $$\Phi^{-1}(\alpha) = \Phi_1^{-1}(\alpha) + \Phi_2^{-1}(\alpha) = 2\Phi_1^{-1}(\alpha),$$ which could only be true if $X \sim N(0, 4)$.