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Transitive Property?

On television recently, the TBS station carried a commercial for the show Everybody Loves Raymond. And I quote:

"If Everybody Loves Raymond
and Raymond is on Wednesdays,
THEN Everybody Loves Wednesdays!
This is an example of the transitive property."

Though admiring an effort to include mathematics as part of this commercial, I question the correctness of its use of mathematics. Is it actually an example of the Transitive Property: If aRb and bRc, then aRc....where a, b, c are statements and R stands for "is Related to"?

Or, is it an example of logic ala Aristotle and his infamous "bArbArA" syllogism:

If A is predicated of all B
and B is predicated of all C,
then A is predicated of all C.

And now a new question arises for you to pursue. Why is it called the "bArbArA" syllogism?