I am trying to solve the following problem:
Let $\mathcal{T}_1, \mathcal{T}_2$ be recursively enumerable $\lambda$-theories such that $\mathcal{T}_1 \subsetneq \mathcal{T}_2$. Show that there is a recursively enumerable $\lambda$-theory $\mathcal{T}_3$ such that $\mathcal{T}_1 \subsetneq \mathcal{T}_3 \subsetneq \mathcal{T}_2$.
I know that if $\mathcal{T}$ is a recursively enumerable $\lambda$-theory such that $\mathcal{T} \nvdash_\lambda M = N$ (for some $\lambda$-terms $M, N$) then there is a term $P$ such that for all term $Q$, $\mathcal{T} \cup \{P = Q\} \nvdash_\lambda M = N$. I am trying to use this fact but to no avail so far. Any hints would be appreciated.