I'm trying to prove the following proposition.
Let $\mathcal{L}$ be a first order language and assume $\mathfrak{A}$ and $\mathfrak{B}$ are two $\mathcal{L}$ -structures. We say $\mathfrak{A}$ and $\mathfrak{B}$ are isomorphic if there is a bijection $F: A \rightarrow B(A$ is the domain of $\mathfrak{A}$ and $B$ is the domain of $\mathfrak{B}$ ) such that
- $F\left(c^{\mathfrak{A}}\right)=c^{\mathfrak{B}}$ for every constant $c \in \mathcal{L}$;
- $F\left(f^{\mathfrak{A}}\left(a_{1}, \cdots, a_{m}\right)\right)=f^{\mathfrak{B}}\left(F\left(a_{1}\right), \cdots, F\left(a_{m}\right)\right)$ for all $m$ -ary function $f$ in $\mathcal{L}$;
- $R^{\mathfrak{A}}\left(a_{1}, \cdots, a_{n}\right)$ if and only if $R^{\mathfrak{B}}\left(F\left(a_{1}\right), \cdots, F\left(a_{n}\right)\right)$ for all $n$ -ary relation $R$ in $\mathcal{L}$. Assume $\mathfrak{A}$ and $\mathfrak{B}$ are isomorphic and $F: A \rightarrow B$ is a bijection that satisfies conditions above. Show for every $\mathfrak{A}$ -assignment $\beta$ and $\mathcal{L}$ -term $t$ and $\mathcal{L}$ -formula:
$\bullet F\left(t^{\mathfrak{A}}[\beta]\right)=t^{\mathfrak{B}}[F \circ \beta]$
$\bullet \mathfrak{A} \models \phi[\beta]$ if and only if $\mathfrak{B} \models \phi[F \circ \beta]$.
Using that $\lnot, \land$ are adequate, I have managed to prove using induction that both of those claims for all but one, $\lnot$. That is, let $\phi = \lnot \phi_1$, where $\mathfrak{A} \models \phi_1[\beta]$ if and only if $\mathfrak{B} \models \phi_1[F \circ \beta]$, for an arbitrarily fixed valuation $\beta$. (That is the induction hypothesis, right?) Then I want to show that $\mathfrak{A} \models \phi[\beta]$ if and only if $\mathfrak{B} \models \phi[F \circ \beta]$. How can I do this?
My problem was that in the other cases, unravelling the definitions gave me something easy to work with, that looked very similar to the induction hypothesis. But in this case, $\mathfrak{A} \models \phi[\beta]$ $\iff$ $\mathfrak{A} \models \lnot \phi_1[\beta]$ $\iff$ $v^{\beta} (\phi_1) = 0$....? So I first thought of defining some sort of strange valuation $\overline{\beta}$ such that $\overline{\beta}(\varphi) = 1 - \beta(\varphi)$, but then realized that this was not even a valuation. So now I'm totally lost. Could anyone help?