I am learning application of the back and forth method, but am not understanding how it is derived? The statement I have is this:
Theory $T$ is complete and has quantifier elimination when we have following for structures $\mathcal{M,N}$ where $\mathcal{N,M}\models T$ and are $\aleph_{0}$-saturated.
i)$\mathcal{M}$ and $\mathcal{N}$ are having the same quantifier free theory.
ii) If we have tuples $a\in M^{n}$ and $b\in N^{n}$ and $k\in M$ with the quantifier free theory of $\mathcal{M}$ over $a$ is equal to the one for $\mathcal{N}$ over $b$, then we have a $l\in N$ so that quantifier free theory of $\mathcal{M}$ over $ac$ is equal to the one for $\mathcal{N}$ over $bl$.
So I can understand why the completeness, but I don't see how we get quantifier elimination. I don't see how having the same quantifier free theories imply full elimination of all quantifiers. Please give advise.