Questions 9-10 Post-Write

9.Statement: ⊢∀x∀y(x=y→y=x)
Proof:
1. {y=x, x}⊢x by (As)
2. {y=x, x}⊢y=x by (As)
3. {y=x, y}⊢y by (As)
4. {y=x, y}⊢y=x by (As)
5. {y=x, x}={y=x, y}⊢y=x by =I from 2 and 4
6. x=y⊢y=x from 5 by (As)
7. ⊢(x=y→y=x) from 6 by →I
8. ⊢∀y(x=y→y=x) from 7 by ∀I
9. ⊢∀x∀y(x=y→y=x) from 8 by ∀I

10. Statement: ⊢∃x(x=x)
Proof:
1. ⊢(x=x) by =I on an atomic formula
2. φ: (x=x)
3. φ [y|x]: (y=y)
4. ⊢∃x(x=x) by ∃I from 1 as φ[y|x] holds from 2 and 3
Statement: ∃x, y((x≠y))
Proof: Suppose that x and y are both sets with one element, z. ∃x, y((z≠z)) is not true. Hence, we cannot definitively prove that there exist at least two things.

Comments - Andrew Furash
It's not clear if x and y are variables or wffs. You can't prove 'x' is x is a variable. Your use of (As) in 6 also does not make sense. The end of your proof is correct, however.
Nice work on 10

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