Theorem sb8eu

Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on [ax-13]. For a version requiring more disjoint variables, but fewer axioms, see sb8euv. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) (New usage is discouraged.)

Hypothesis
sb8eu.1
\( \vdash \Finv y \, \varphi \)
Conclusion
\( \vdash \exists{!} x \, \varphi \leftrightarrow \exists{!} y \, [ y / x ] \, \varphi \)
Proof
sb8eu.1
nfsb _
\( \vdash \Finv y \, [ w / x ] \, \varphi \)
sb8eulem _