Proof: Sum of minimum of two functions vs minimum of sum of functions

I wonder whether the following is true:

$$\min\big\{f_1(x),g_1(x)\big\} + \min\big\{f_2(x),g_2(x)\big\}\le \min\big\{f_1(x)+f_2(x),~g_1(x) + g_2(x)\big\}.$$

I already know how to prove that:

$$\min\{f(x)\}+\min\{g(x)\} \le \min\{f(x)+g(x)\},$$

but I cannot proceed with the handling of the minimum of two functions.