Theorem eubid

Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 19-Feb-2023.)

Hypotheses
eubid.1
\( \vdash \Finv x \, \varphi \)
eubid.2
\( \vdash \varphi \rightarrow (\psi \leftrightarrow \chi) \)
Conclusion
\( \vdash \varphi \rightarrow (\exists{!} x \, \psi \leftrightarrow \exists{!} x \, \chi) \)
Proof
l0
eubid.1
eubid.2
l2
alrimi l0 _
\( \vdash \varphi \rightarrow \forall x \, (\psi \leftrightarrow \chi) \)
eubi
\( \vdash \forall x \, (\psi \leftrightarrow \chi) \rightarrow (\exists{!} x \, \psi \leftrightarrow \exists{!} x \, \chi) \)
syl l2 _