My deductive argument against orthodox (no capital O) trinitarianism

Deductive argument against the trinity
AXIOMS
1. A = YHWH
2. X = The Father
3. Y = The son
4. X = A ∨ X ≠ A
5. Y = A ∨ Y ≠ A

PREMISES
1. X = A
2. Y = A
3. Y ≠ X
∴ ⊥

Into simple words

AXIOMS/Truths presupposed for the argument
1. Let A be YHWH/God
2. Let X be God The Father
3. Let Y be God the Son
4. X either is A or isn't A
5. Y either is A or isn't A

Premises/Argument
1. X is A
2. Y is A
3. Y is not X

Therefore, this is a contradiction. Because X and Y are both A, but not eachother.
Further explination: If the Father and Son are fully God, they must be identical to eachother. Why? Because they are both God. Since something is or is-not God, something that is-God is fully God with no distinction. If one of them are "identical/alike" to God, then they are not God. But since, according to orthodox trinitarianism, they are numerically distinct, this posits a contradiction under their own logic and semantics.

AXIOMS
1. A = YHWH/God/Divine essence
2. X = The Father
3. Y = The son
4. (X = A) ∨ (X ≠ A)
5. (Y = A) ∨ (Y ≠ A)
6: T = Trinity
7. M = Modalism
8. (M = T) → ⊥
9. (M ≠ T) → ¬⊥

(X = A) ∧ (Y = A) → (X=Y)

Simplified:
X is A, and Y is A, therefore, X is Y.

In Trinitarianism: X (The Father) and Y (The Son) are not identical, therefore,

(X = A) ∧ (Y = A) ∧ (X ≠ Y) → ⊥

Simplified: X is A, and Y is A. X is not Y. Therefore, contradiction.

Trinitarian counterargument: "The persons are numerically distinct from eachother, and count as something different apart of divine essence."

Reductio ad absurdum with 2 possibilities:

(1) "The persons and the essence are equal and that is no distinction, aka the essence is the persons.":

(a) (X = A) ∧ (Y = A) ∧ (X=Y) → ¬⊥M.
(b) (X = A) ∧ (Y = A) ∧ (X=Y) → ⊥T.

Simplified:
(a) X is A, and Y is A, and X is Y. Therefore, no contradiction to modalism.
(b) X is A, and Y is A, and X is Y. Therefore, contradiction to Trinitarianism. (The person isnt the divine essence)
∴ This collapses into modalism and contradicts Trinitarianism, because the persons are identical. Therefore, Trinitarianism is false.

(2) "The persons are distinct from the divine essence aka the essence is-not the persons."

Then the argument stands, because:
(a) (X = A) ∨ (X ≠ A)
(b) (X ∼ A) → (X ≠ A)
∴(X ≠ A) → ⊥T

Simplified:

(a) Law of non-contradiction (X is either A, or Not-A)
(b) If X is not fully identical/only alike to A, then, X is not-A. (The person isnt the divine essence)
∴ X is not A, therefore the trinity is contradictory

Did you articulate an orthodox conception of the trinity? Since different Christian doctrines hold different understandings of the trinity, it doesn't make sense to treat them the same and so I will focus on Orthodox (uppercase O) trinitarianism. Nicaea's formulation of it is something like: There is one divine essence/substance (ousia), and three distinct persons (hypostases). For Nicaea, that the Son is God is a short-hand way of saying that the Son is of the same substance as God. With this in mind, your premises strawman worthwhile understandings of trinitarianism, including that for Orthodox Christians who want to thread a needle between the Son being God, and the Son being a participation in God's essence.

If Fido is a dog, and Rex is a dog, that doesn't make Fido equal to Rex. And it doesn't even make Fido equal to a dog, nor does it make Rex equal to a dog. Even if the trinitarian holds that the Father is fully and completely God, it doesn't make the Son equal to the Father nor does it make the Son equal to God. It might (or might not necessarily) make the Father equal to God, but that alone wouldn't be enough to generate an internal contradiction.

Since you are using "=" to mean at least two different things simultaneously, your formalization fails before the theology even enters. Arguments need numerical identity throughout to generate a contradiction. One of your "axioms" (which is even a misuse of what the word axiom means) can be articulated as "There can be at most one thing that fully instantiates A." which is a substantive metaphysical claim that Nicene trinitarianism explicitly denies.

(1) this is a universal conception of the trinity, all trinitarian conceptions concede:
all the 3 persons are god, they are not eachother

(2) youre equivocating universals vs formal identity. my argument is from formal identity, not universals.
if A is B and B is C then A is C. this would not be true with a universal, however this isnt my argument.

(3) the argument itself clarifies "=" means formal identity. i actually think you dont understand the argument or ignore its basis

Was the trinity concept ever supposed to make strict logical sense?

1. The trinity concedes that all three Persons are God and not each other. But this could mean so many different things, it could mean at least thousands of different things.
2. If a formal identity means a strict identity, then you are strawmanning the trinity. If a formal identity means something less strict than strict identity, then a contradiction doesn't follow from your conception of the trinity.

Some doctrines believe in making logical sense of the trinity, and others don't. https://plato.stanford.edu/entries/trinity/

The whole concept of the trinity was stupid and unnecessary from the beginning. There is God the Father, of whom Jesus the man was some form of incarnation. Simple.