Your situation is too involved for an online answer.
Using the written documentation you have is a good place to begin. Contact a local lawyer - many may give you a consultation - to discuss your specifics. Far too many variables exist in the short post you wrote for any further observation by me.
You might find my Legal Guide helpful "How to Choose A Lawyer For You"
You might find my Legal Guide helpful " What Do I Tell My Lawyer"
No one can know what the record is in the case because online we cannot see your documents. You need a lawyer. Check with a lawyer in your locale to discuss more of the details.
Good luck to you. God bless. Best of luck to you.
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See an attorney, but keep in mind that C (unless he has collected the judgment) may have a defense that the lack of the condition in the judgment is a "scrivener's error" that can be corrected "nunc pro tunc" (which literally means "now for then") to reflect the "true" judgment of the court. The Items which might support that position could be hearing transcripts (if any) and the moving papers and pleadings. And, the possibility of C and B supporting this action vs A, as B does not want to come up with the money.
If this "fix" is not made, generally the "collateral effect" extends only to issues "actually litigated." If, for example, the issue was properly framed as a conditional argument (only actually litigating "if C is liable to A, what happens?" as opposed to "whether C is liable to A") then "whether C is liable to A" was, arguably, not "actually litigated." From the unconditional judgment language, though, it could also be argued that this question was "necessarily decided."
Also, the fact that the litigation is between different parties (C vs B, not A vs C) may preclude the collateral effect. Finally, given that the judgment is two years old and the parties have continued to litigate as if the it was framed as conditional, the judge may deem the argument waived or barred for equitable reasons. That is, why did A waste everybody's time if the case was won two years ago?
"Judicial admissions" may definitely be used regardless of any of the above, but they need to be relatively unequivocal. So, if C claimed "I am liable to A" as opposed to "I may be liable to A" then these admissions would be admissible to prove A's case." Understand though, that the admissions will be subject to context that can be used to show they are not what they appear.
Without having all the details my "gut feeling" is that if the judgment has not been collected and all the parties have treated the judgment as if it was conditional for the last two years of litigation, your theory might not fly. On the other hand, if C has collected, or attempted to collect the entire judgment from B, then you are right and A should win. I suspect this is not the case, however, as B would have raised the issue that the judgment was "incorrect" if collection attempts were made. Instead, I would expect that C has more or less washed his hands of the litigation and is letting B do the heavy lifting defending the case.
So, if your question is, "is this a slam dunk?" my answer is probably not, at least without more detail. Is it worth exploring whether the circumstances are such that this argument would provide a quick route to victory or be a sufficiently strong argument to force a favorable settlement? Absolutely.
Many lawyers have never had the need to become well-versed in these questions, and the courts have spilled oceans of ink discussing the intricacies of these issues. Good for you for spotting the argument. Have you considered law school?