Metamath Proof Explorer


Theorem eubidv

Description: 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)

Ref Expression
Hypothesis eubidv.1
|- ( ph -> ( ps <-> ch ) )
Assertion eubidv
|- ( ph -> ( E! x ps <-> E! x ch ) )

Proof

Step Hyp Ref Expression
1 eubidv.1
 |-  ( ph -> ( ps <-> ch ) )
2 1 alrimiv
 |-  ( ph -> A. x ( ps <-> ch ) )
3 eubi
 |-  ( A. x ( ps <-> ch ) -> ( E! x ps <-> E! x ch ) )
4 2 3 syl
 |-  ( ph -> ( E! x ps <-> E! x ch ) )