My friend is telling me that fishing for salmon east of the Antioch bridge is illegal.  I cannot find a restriction anywhere for that area.  Anyone...?

your friend is correct. east of the antioch bridge is considered the San Joaquin, where salmon can't be kept.

7.50 Alphabetical Waters with special regs

156.5 Sacramento River
(E)Sacramento River from the Hwy 113 bridge near Knights Landing to the
Carquinez Bridge (includes Suisun Bay, Grizzly Bay and all tributary sloughs
west of Highway 160).

San Joaquin River does not explicitly allow you to keep salmon, so you can't.

anyways, the sac river definition includes