Per the FAR's yes... ( I assume when you said Dual Given right after your private you mean Dual Received) ... however the airlines see it differently they only count PIC time as time that you didn't have an instructor on board (AKA Your the sole manipulator of the controls/ Final authority). Its the same thing with dual Given, even though you are in fact the PIC when you are flight instructing( because you are the final authority) Airlineapps.com will add this up separately from PIC time. Because some airlines actually value (or used to) Flight Instructor time over the I got a 100 dollar burger PIC time people log.

This all used to matter because when airlines weren't hiring at next to nothing times they had a certain amount of PIC requirements, and anything where you are logging DUAL RECEIVED would not count towards that.

You will find that at most flight schools that are oriented towards people that will eventually end up at Airlines (I.E. Aviation Universities, 0 to Jet schools, most other 141 operations) students are usually told not to log Dual received time as PIC time because even though it is legal these schools now know how future employers feel about the issue.