Law of Syllogism

If [p → q] and [q → r] are both true,
then [p → r] is true.

This is the law of syllogism.

Conditional Statement

The first statement is a conditional statement.

So the statement behind if is p:
it's raining.

And the statement behind then is q:
I'm staying home.

So the first statement is p → q.

The second statement is also
a conditional statement.

The statement behind if is q.

And the statement behind then is r:
I'm listening to music.

So the second statement is q → r.

It says
both [p → q] and [q → r] are true.

Then, by the law of syllogism,
[p → r] is true.

Write p → r.

If, p, it's raining,

then, r, I'm listening to music.

So

If it's raining,
then I'm listening to music

is always true.