If I have
$$ \forall x: (x < 5) \rightarrow (f(x)<g(x)), $$ $$ \forall x: (x < 5) \rightarrow (g(x)<h(x)) $$
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.
What Wiki calls the Composition Rule seems to be the one that combines implications:
$$ ((p \rightarrow q)\land (p\to r)) \Leftrightarrow (p \rightarrow (q\land r)) $$
So would the correct order here be Universal Instantiation (aka Universal Elimination), Composition, then Universal Generalization (aka Universal Introduction)? Not sure