Theorem reubidva

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004) Reduce axiom usage. (Revised by Wolf Lammen, 14-Jan-2023)

	Ref	Expression
Hypothesis	reubidva.1	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
Assertion	reubidva	|- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

Proof

Step	Hyp	Ref	Expression
1		reubidva.1	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
2	1	pm5.32da	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
3	2	eubidv	|- ( ph -> ( E! x ( x e. A /\ ps ) <-> E! x ( x e. A /\ ch ) ) )
4		df-reu	|- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
5		df-reu	|- ( E! x e. A ch <-> E! x ( x e. A /\ ch ) )
6	3 4 5	3bitr4g	|- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )