[proofplan]
We prove that every formula with parameters from $A$ has the same truth value on $b$ in $M$ and on $c$ in $N$. Quantifier elimination replaces the formula uniformly, before parameters are substituted, by a quantifier-free formula modulo $T$. Since both $M$ and $N$ are models of $T$, the original formula and its quantifier-free replacement agree in both structures. The hypothesis on quantifier-free types then transfers truth from $M$ to $N$, and this is exactly equality of complete types over $A$.
[/proofplan]
[step:Replace an arbitrary formula by a quantifier-free formula uniformly in its parameters]
Let $L$ denote the first-order language of the theory $T$. Fix $m \in \mathbb{N}$, an $L$-formula $\varphi(x_1,\dots,x_n,y_1,\dots,y_m)$, and a tuple $a=(a_1,\dots,a_m) \in A^m$. Since $T$ has quantifier elimination, there exists a quantifier-free $L$-formula $\psi(x_1,\dots,x_n,y_1,\dots,y_m)$ such that
\begin{align*}
T \models \forall x_1 \cdots \forall x_n \forall y_1 \cdots \forall y_m \bigl(\varphi(x_1,\dots,x_n,y_1,\dots,y_m) \leftrightarrow \psi(x_1,\dots,x_n,y_1,\dots,y_m)\bigr).
\end{align*}
Because $M \models T$ and $N \models T$, this equivalence holds in both structures. Therefore
\begin{align*}
M \models \varphi(b,i_M(a)) \quad &\Longleftrightarrow \quad M \models \psi(b,i_M(a)),\\
N \models \varphi(c,i_N(a)) \quad &\Longleftrightarrow \quad N \models \psi(c,i_N(a)).
\end{align*}
[guided]
We begin with an arbitrary formula, because equality of complete types means agreement on every formula over the parameter set. Thus fix $m \in \mathbb{N}$, an $L$-formula
\begin{align*}
\varphi(x_1,\dots,x_n,y_1,\dots,y_m),
\end{align*}
and a parameter tuple $a=(a_1,\dots,a_m) \in A^m$. The tuple $i_M(a)=(i_M(a_1),\dots,i_M(a_m))$ is the interpretation of these parameters in $M$, and $i_N(a)=(i_N(a_1),\dots,i_N(a_m))$ is the corresponding interpretation in $N$.
The key point is that quantifier elimination is applied before substituting parameters. Since $T$ has quantifier elimination, there is a quantifier-free $L$-formula
\begin{align*}
\psi(x_1,\dots,x_n,y_1,\dots,y_m)
\end{align*}
with the same free variables as $\varphi$ such that $T$ proves their universal equivalence:
\begin{align*}
T \models \forall x_1 \cdots \forall x_n \forall y_1 \cdots \forall y_m \bigl(\varphi(x_1,\dots,x_n,y_1,\dots,y_m) \leftrightarrow \psi(x_1,\dots,x_n,y_1,\dots,y_m)\bigr).
\end{align*}
This uniformity matters: the same quantifier-free formula $\psi$ works for every possible substitution of the parameter variables $y_1,\dots,y_m$.
Now use that both structures are models of $T$. Since $M \models T$, the displayed equivalence is true in $M$, and substituting $b \in M^n$ for the variables $x_1,\dots,x_n$ and $i_M(a) \in M^m$ for the variables $y_1,\dots,y_m$ gives
\begin{align*}
M \models \varphi(b,i_M(a)) \quad \Longleftrightarrow \quad M \models \psi(b,i_M(a)).
\end{align*}
Likewise, since $N \models T$, substituting $c \in N^n$ and $i_N(a) \in N^m$ gives
\begin{align*}
N \models \varphi(c,i_N(a)) \quad \Longleftrightarrow \quad N \models \psi(c,i_N(a)).
\end{align*}
Thus the original formula has been reduced, in both models and for the same parameter tuple, to the quantifier-free formula $\psi$.
[/guided]
[/step]
[step:Transfer truth of the quantifier-free replacement using the hypothesis]
The formula $\psi(x_1,\dots,x_n,y_1,\dots,y_m)$ is quantifier-free. By the hypothesis that $b$ and $c$ have the same quantifier-free type over the identified copy of $A$,
\begin{align*}
M \models \psi(b,i_M(a)) \quad \Longleftrightarrow \quad N \models \psi(c,i_N(a)).
\end{align*}
Combining this equivalence with the two equivalences from the previous step yields
\begin{align*}
M \models \varphi(b,i_M(a)) \quad \Longleftrightarrow \quad N \models \varphi(c,i_N(a)).
\end{align*}
[/step]
[step:Conclude equality of complete types over $A$]
The formula $\varphi(x_1,\dots,x_n,y_1,\dots,y_m)$, the integer $m \in \mathbb{N}$, and the parameter tuple $a \in A^m$ were arbitrary. Hence $M$ and $N$ agree on the truth value of every $L$-formula with free object variables $x_1,\dots,x_n$ and parameters from the interpreted copies $i_M(A) \subset M$ and $i_N(A) \subset N$ when evaluated at $b$ and $c$, respectively. By the definition of equality of complete types over a parameter set, this is precisely the assertion that $b$ and $c$ have the same complete type over $A$.
[/step]
