Okay, I completely fail to follow your train of thought. Let me see if I can work it out with an example based on OP's question:

to believe something without evidence simply because you believe something related to it that said belief would support

I believe that all members of the Example Party of Canada (EPC) are gullible simply because I believe James is a gullible person and James is a supporter of the EPC.

in the absence of some indepently verifiable proof P of fact F, where F supports some argument A, believing F => A is still logically invalid till you have P such that P => F

Fact F: All members of the EPC are gullible. For all x ∈ EPC, gullible(x).

Argument A: James is gullible. gullible(j); j ∈ EPC.

Proof P: A rigorous study that shows that every single member of the EPC is in fact gullible, QED. This proof does not exist, but if it did, P would indeed prove F.

Statement F=>A is in fact mathematically valid and correct regardless of the value of F. But the biased opinion is F, and not F=>A.

The reason just assuming F => A is unsound is that you don't know if there exists some T such that T => ~F

Proof T: John, a member of the EPC, is not gullible, as determined by a thorough and objective study. But you don't know that. Naturally, T=>~F. In other words, we have determined that there exists y ∈ EPC such that ~gullible(y), QED.

which would obviously invalidate the belief that F=>A

However, F=>A is still true and valid!

TL;HM

We know A and we know F=>A; we know that A ^ (F=>A) do not imply F. The whole argument above is simply the following: A does not imply F; you cannot assume F is true unless you have P because until you have P you cannot be certain that T is the case instead.