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given problem is, If $\Sigma\vdash\varphi$ iff $\Sigma\vdash\psi$, then $\Sigma\vdash\varphi\leftrightarrow\psi$.
I can prove this using sound&completeness theorem but I don't know how to do without those theorems.
without them, I proved when $\Sigma\vdash\varphi$ is true, but I couldn't prove when $\Sigma\nvdash\varphi$. without the theorems, I don't know how $\nvdash$ part contributes to prove the problem.
p.s. only member of given set, tautology, and Modus Ponens are allowed for deduction.