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Given the axioms below and the rules of Modus Ponens and Universal Generalization, how can you prove that t=s → s=t for any terms s and t? Additionally, how do you prove that t = s → (s = r → t = r) ?

Axioms:

1) ϕ → (Ψ→ϕ )

2) (ϕ → (Ψ→χ)) → ((ϕ→Ψ) → (ϕ→χ)

3) (~Ψ→~ϕ) → ((~Ψ→ϕ) →Ψ)

4) ∀αϕ → ϕ(β/α)

5) ∀α(ϕ→Ψ) → (ϕ→∀αΨ)

6) ∀x(x=x)

7) x=y → (ϕ(x,x) → ϕ(x,y))

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    Hi, welcome to Philosophy SE. This looks like a HW problem. Please provide the source you got the axioms from (the text for your class, I am guessing), and explain how you approached solving it. – Conifold Apr 2 '17 at 22:18
  • You can find it in every math log textbook. Use Ax.7 with a suitable instance of ϕ. – Mauro ALLEGRANZA Apr 3 '17 at 6:06
  • You can see this post for a proof. – Mauro ALLEGRANZA Apr 3 '17 at 6:40

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