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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.

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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

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