Brainstorm debuted in
Justice League of America #32 (December 1964) fighting Superman, Batman, Flash, Atom, and Hawkman, but he was really after Green Lantern. He returned in #36 (June 1965) fighting the whole JLA.
AFAIK, his next appearance was in
Superman #294 (December 1975) where he fought the Man of Tomorrow one-on-one. I don't know of any appearances after that. Too bad. He made a "good" villain.
...the Justice League fought their own villains like Doctor Light and Felix Faust for YEARS before a bad guy from one of the solo member's books popped in...
As soon as I read this, I was sure I could disprove your claim. However, the only evidence I could uncover was...
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JLA #21-22, where Mr. Element (a Flash-rogue) and Chronos (an Atom-opponent) teamed with Felix Faust and three villains from Earth-Two (the Wizard, the Fiddler, and the Icicle -- old Injustice Society pals) as the Crime-Champions.
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JLA #34, where the Joker (you-know-who's foe) and Chac (who previously fought Hawkman) appeared together. But this doesn't count because they were really just dream creations of Doctor Destiny!
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JLA #61, which ended with a battle royale involving several of the JLAer's individual enemies (the Penguin, Lex Luthor, Tattooed Man, Captain Boomerang, I.Q., Jason Woodrue, etc.). But once again Doctor Destiny was behind the plot, and the other villains seemed to be in his thrall!
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JLA #77, where the Joker tricked Snapper Carr in a plot to turn public opinion against the JLA. And no Doc Destiny in sight! Finally, one member's foe acting alone and in his right mind (well, as "right" as the Joker's mind can get) took on the whole team.
Okay. That last one was from 1969, nine YEARS :shock: after the JLA's debut. :oops: I yield to your superior knowledge, Julian.