Inference rule's restriction

Stream: maths

[Adrian Bono]

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As you see above, the second rule has two restrictions. x should not be free in Γ or Psi, or term should not occur in these two. Can someone show that if term occurs in psi and Γ , the rule would be unsound?
similar restriction for generalization theorem is proved in the way below, but I could not prove the similar notion for Introduction of ∃
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