Replace fire bricks in fireplace

I have a masonry fireplace where about 8 fire bricks have fallen off of the back wall of the firebox. The fireplace was built in the late 1960’s, then mostly unused for decades. I was told that the last time it was used was around 2005. The room was un-climate controlled for long periods and could have had lots of moisture exposure, but is in Texas so unlikely to have experienced freeze spalling.

The bricks do not appear to be keyed in any way, just mortared in with mortar that failed.

Can I remove all of the loose fire brick, then reinstall them with a proper refractory mortar, or is there something different that should be done? Do I mortar the fire brick only to fire brick, or also to the regular brick wall behind it?