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 ) )