Theorem reueubd 3164

Description: Restricted existential uniqueness is equivalent with existential uniqueness if the unique element is in the restricting class. (Contributed by AV, 4-Jan-2021.)

Hypothesis

Ref	Expression
reueubd.1	|- ( ( ph /\ ps ) -> x e. V )

Assertion

Ref	Expression
reueubd	|- ( ph -> ( E! x e. V ps <-> E! x ps ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   V( x)

Proof of Theorem reueubd

Step	Hyp	Ref	Expression
1		reueubd.1	. . . . 5 |- ( ( ph /\ ps ) -> x e. V )
2	1	ex 450	. . . 4 |- ( ph -> ( ps -> x e. V ) )
3	2	pm4.71rd 667	. . 3 |- ( ph -> ( ps <-> ( x e. V /\ ps ) ) )
4	3	eubidv 2490	. 2 |- ( ph -> ( E! x ps <-> E! x ( x e. V /\ ps ) ) )
5		df-reu 2919	. 2 |- ( E! x e. V ps <-> E! x ( x e. V /\ ps ) )
6	4, 5	syl6rbbr 279	1 |- ( ph -> ( E! x e. V ps <-> E! x ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   E!weu 2470   E!wreu 2914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-reu 2919

This theorem is referenced by:  frgreu  27132