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Marker Theorem 3.1.4 says: Suppose T $T$ is a theory in a language with at least one constant symbol. Then an L-formula $L$-formula $\phi(x)$ is T-equivalent $T$-equivalent to a quantifier-free formula iff, whenever M $M$ and N $N$ are models of T, $T$, $A \subseteq M$ and $A \subseteq N$, then $M |= \models \phi(a)$ iff $N |= \models \phi(a)$ for any $a$ from $A$. -end of theorem

Thus, if we further assume that $T$ is model-complete, then it must follow that $T$ eliminates quantifiers? Where am I going wrong, if it is wrong?

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# Does model-complete in a language with a constant symbol imply EQ?

Marker Theorem 3.1.4 says: Suppose T is a theory in a language with at least one constant symbol. Then an L-formula $\phi(x)$ is T-equivalent to a quantifier-free formula iff, whenever M and N are models of T, $A \subseteq M$ and $A \subseteq N$, then $M |= \phi(a)$ iff $N |= \phi(a)$ for any $a$ from $A$. -end of theorem

Thus, if we further assume that $T$ is model-complete, then it must follow that $T$ eliminates quantifiers? Where am I going wrong, if it is wrong?