Lois, the monsoons typically start the second or third week of July, not the first week. I have... DECADES of experience with 'em.
You'll know when it's been raining in roughly the red circle below for a week or two. Until that happens, no monsoon. I've seen it as early as mid-June, as late as the third week of August. Also a couple of years with little or no monsoon activity the whole summer.
That area of Mexico is where all of the humidity for the monsoons comes from, so it'll always lead the AZ monsoons by a week plus.
Unfortunately, the week BEFORE monsoons begin in Tucson & Phoenix is the hottest, nastiest fucking week in the whole year. Asphalt melts at mid-July temps. If your air conditioner is gonna die, it'll die in July 'cos it's running 24/7. The
fucking weather in AZ *sucks* in July.
OBTW, it's only been mid-90s here in Dallas!