Enderton 2.5 problem 1.

(Semantic rule EI) Assume that the constant symbol $\displaystyle c$ does not occur in $\displaystyle \phi, \psi$, or $\displaystyle \Gamma$, and that $\displaystyle \Gamma;\phi_c^x \models \psi$. Show (without using the soundness and completeness theorems) that $\displaystyle \Gamma; \exists x \phi \models \psi$.

If I were able to use the soundness theorem, I could use the rule EI and apply the soundness theorem. However, it is not allowed to use the soundness theorem.

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Let A be a structure for the language and let $\displaystyle s:V \rightarrow |A|$ such that if $\displaystyle \models_A \phi_c^x[s]$, then $\displaystyle \models_A \psi[s] $. Now it suffices to show that $\displaystyle \models_A \exists x \phi[s]$.

I can't proceed from here.

Any help will be appreciated.