Theorem mobidv

Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.)

Hypothesis
mobidv.1
\( \vdash \varphi \rightarrow (\psi \leftrightarrow \chi) \)
Conclusion
\( \vdash \varphi \rightarrow (\exists^\ast x \, \psi \leftrightarrow \exists^\ast x \, \chi) \)
Proof
mobidv.1
l1
alrimiv _
\( \vdash \varphi \rightarrow \forall x \, (\psi \leftrightarrow \chi) \)
mobi
\( \vdash \forall x \, (\psi \leftrightarrow \chi) \rightarrow (\exists^\ast x \, \psi \leftrightarrow \exists^\ast x \, \chi) \)
syl l1 _