Theorem mobid

Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on [ax-10], [ax-11], [ax-13]. (Revised by BJ, 14-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)

Hypotheses
mobid.1
\( \vdash \Finv x \, \varphi \)
mobid.2
\( \vdash \varphi \rightarrow (\psi \leftrightarrow \chi) \)
Conclusion
\( \vdash \varphi \rightarrow (\exists^\ast x \, \psi \leftrightarrow \exists^\ast x \, \chi) \)
Proof
l0
mobid.1
mobid.2
l2
alrimi l0 _
\( \vdash \varphi \rightarrow \forall x \, (\psi \leftrightarrow \chi) \)
mobi
\( \vdash \forall x \, (\psi \leftrightarrow \chi) \rightarrow (\exists^\ast x \, \psi \leftrightarrow \exists^\ast x \, \chi) \)
syl l2 _