Metamath Proof Explorer


Theorem sb8eu

Description: 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.)

Ref Expression
Hypothesis sb8eu.1
|- F/ y ph
Assertion sb8eu
|- ( E! x ph <-> E! y [ y / x ] ph )

Proof

Step Hyp Ref Expression
1 sb8eu.1
 |-  F/ y ph
2 1 nfsb
 |-  F/ y [ w / x ] ph
3 2 sb8eulem
 |-  ( E! x ph <-> E! y [ y / x ] ph )