有没有更好的方法来对齐这个证明的各个部分？

Question

有没有更好的方法来对齐这个证明的各个部分。

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\begin{document}
Notice that
    \begin{align*}
        (a_{0}, b_{0}) \equiv (a_{1}, b_{1}) &\hspace{4pt} \text{iff} \quad \text{definition of } \equiv \\
        \begin{aligned} a_{0} \neg (a_{0} \wedge b_{0}) &= a_{1} \neg (a_{1} \wedge b_{1}) \text{ and } \\
        b_{0} \neg (a_{0} \wedge b_{0}) &= b_{1} \neg (a_{0} \wedge b_{0}) \end{aligned} \bigg\} &\hspace{4pt} \text{iff} \quad \text{using proposition 11.1} \\
        \begin{aligned} a_{0} \wedge b_{0}^\prime &= a_{1} \wedge b_{1}^\prime \text{ and } \\ b_{0} \wedge a_{0}^\prime &= b_{1} \wedge a_{1}^\prime \end{aligned} \bigg\} &\hspace{4pt} \text{.}
    \end{align*}
    The first statement holds.
\end{document}