Theorem eubidv

Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.)

Hypothesis
eubidv.1
\( \vdash \varphi \rightarrow (\psi \leftrightarrow \chi) \)
Conclusion
\( \vdash \varphi \rightarrow (\exists{!} x \, \psi \leftrightarrow \exists{!} x \, \chi) \)
Proof
eubidv.1
l1
alrimiv _
\( \vdash \varphi \rightarrow \forall x \, (\psi \leftrightarrow \chi) \)
eubi
\( \vdash \forall x \, (\psi \leftrightarrow \chi) \rightarrow (\exists{!} x \, \psi \leftrightarrow \exists{!} x \, \chi) \)
syl l1 _