Fascinating! Let me try.
I claim that mathematics is subjective. All I have to show is that different kinds of mathematical reasoning involves using different parts of the brain and voila!
But, no, sadly, that's not how it works. We all agree that all of our thinking involves using our brains, and so the fact that moral reasoning is reflected in brain activity is not relevant.
As well, we all agree that some moral claims reflect purely subjective biases. No realist argues that every time a person makes a moral claim, he is doing so free from emotional distortions. Rather, the realist says: regardless of one's reason for believing a certain moral judgment, the judgment itself expresses an objective claim, the sort of thing which is either true or false.
Of course, I am not arguing in favor of realism. Rather, I am evaluating your claim that subjectivism is obviously true. And I'm still waiting for a clear argument to that effect. The fact that moral reasoning involves using the brain is irrelevant, unless you have some remarkable study that objective reasoning doesn't show up in brain scans.
Now it gets tricky.

We have to go all the way back to the 3 classic laws of thought.
- A = A
- ~(A = ~A)
- A OR ~A.
These are in general considered objective as per reasoning. I agree.
Now we reach the limit of my English, because I know I will misuse or get wrong some important word.
For #1 it is in short - What is, is (at a given time and in a given sense).
#2 - What is, can't both be and not be (at a given time and in a given sense).
#3: What is, is either what it is or it is something else (at a given time and in a given sense).
Now in physical terms it is at a given time and place and in a given sense.
Notice this for something to be false/wrong according to this it has take place at the same time and place whether it is so or differently. That is where my English fails me. A contradiction is a claim of something, which is in a given sense or not in a given for exactly the same time and place.
So for A to be false or not false, the law of non-contradiction must kick in, so A must at the same time and place both be A and not A.
Okay, we have ruled out objective through observation. Further we don't accept distortion by personal feelings or prejudices. What is left is without distortion by personal interpretation.
Here it gets funny. Imagine you and I witness a killing. Remember Hume and not objective through observation, so here it goes; me as the realist and you as the anti-realist. B and C are interchangeable as either right or wrong.
Me: The killing(K) was B and not C.
You: That is a personal interpretation.
Me: How do you know that?
You: Because of how cause and effect work. It wasn't caused through observation; i.e. the "was" is not in the killing.
Me: Yes, but it was without personal interpretation.
You: No, because look at K is B and not C as said by you, TJ. The "is" is caused by you. The "is" has its cause in you. So now I say K is C and not B.
Me: That is a contradiction.
You: No. They don't take place at the same time and place. So it is not a contradiction. It seems that way to you, because you reduce away time and place.
Me: That is meaningless.
You: That is a result of your personal interpretation. "K is" is no matter what follows a personal interpretation because it is a given single person, which personally assigns "B and not C" or "C and not B".
Hi phiwum

Let me sum up. For any case of the truth value of a moral claim, the truth value is assigned by personal interpretation, feelings or prejudices. It can't be any other way, unless you want to deny cause and effect.
With regards
PS - I would love for you to explain how it can be done objectively for true and false.
