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.