If A is B, but B is A and C, then assuming that A is not C is it possible that [snip]
That's all you need to show that your antecedent is a contradiction. Convert this into first-order logic, and it's simply a conjunction of equalities. I'll use a plus for conjunction since this board doesn't seem to support LaTeX's \wedge.
(A=B) + (B=A) + (B=C) + (A!=C) -> [anytthing]
The left hand side is clearly a contradiction:
A=B + B=C <=> A=C
which yields A=C + A!=C in the antecedent.
Now, once you have a contradiction as the antecedent of a material impliciation, you can imply anything at all. Remember the truth table for material implication:
A B A->B
T T T
T F F
F T T
F F T
Hence, the consequent is true. The general scientific rule here is that if your assumptions contain a contradiction, you can prove absolutely anything. Insert discussion of evolution vs. creationism here.
The author is probably trying to be clever by inserting an ambiguous "or" in the consequent. If this is inclusive, then we're done. If it is exclusive, then only one of the components of the disjunction can be true. However, once we've shown a contradiction in the antecedent, pragmatic philosophers wouldn't care.
I wouldn't call it armchair philosophy: I would call it a mean homework assignment from a discrete maths course.