# Which logical rules are used in combining universal quantifiers with same conditional

If I have

$$\forall x: (x < 5) \rightarrow (f(x) $$\forall x: (x < 5) \rightarrow (g(x)

I'm allowed to combine the implication into:

$$\forall x: (x < 5) \rightarrow (f(x) < g(x) < h(x))$$

Which logical rule or rules allows me to combine the expressions because they both start with $$\forall x: (x < 5)$$. I suspect that there may be three rules at play here, one to eliminate the quantifiers, another to combine the implications, and another to reintroduce the quantifiers.

$$((p \rightarrow q)\land (p\to r)) \Leftrightarrow (p \rightarrow (q\land r))$$