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