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The premise ∃x∃y∀z(x = z ∨ y = z) says that there exists at most two distinct values in the domain. The conclusion ∀x∀y(¬x = y → ∀z(x = z ∨ y = z)) says that for any two distinct values in the domain, there does not exist a third. Clearly we need to utilise existential elimination and universal introduction. ∃-elim here will be tricky. The 'trick' is ...

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