Expressing a statement in first order logic

I am currently confused, below I have a statement and my expression of that statement in first order logic.

Reference:

W(person, song)

R(person, song, album)

Statement: Every song that Gershwin wrote has been recorded on some album.

Answer: ∀y W(G, y) ⇒ ∃a R(G, y, a)

Answer: ∀y W(G, y) ⇒ ∃a,p R(p, y, a)

Instead of using Gershwin in the argument of R, why did it instead say there exists a person p? Does it really make a difference or am I still unclear about how expressing statements in first order logic work?

• Because $p$ is the person who recorded song $y$. So, if $y$ stands for "It's Wonderful," $p$ could be Barbara Hendricks. – Fabio Somenzi Nov 3 '18 at 5:51
• Indeed, after David pointed it out below I facepalmed myself. – Belphegor Nov 3 '18 at 5:53
• Apparently in R(person, song, album) the variable "person" stands for the recording artist, not for the composer. The textbook answer comes down to: for every song $y$ , if it is composed by Gershwin, you can be certain that it is recorded on an album $a$ by an artist $p$. – M. Wind Nov 3 '18 at 5:53