MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reueubd Structured version   Visualization version   GIF version

Theorem reueubd 3312
Description: Restricted existential uniqueness is equivalent to existential uniqueness if the unique element is in the restricting class. (Contributed by AV, 4-Jan-2021.)
Hypothesis
Ref Expression
reueubd.1 ((𝜑𝜓) → 𝑥𝑉)
Assertion
Ref Expression
reueubd (𝜑 → (∃!𝑥𝑉 𝜓 ↔ ∃!𝑥𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem reueubd
StepHypRef Expression
1 reueubd.1 . . . . 5 ((𝜑𝜓) → 𝑥𝑉)
21ex 401 . . . 4 (𝜑 → (𝜓𝑥𝑉))
32pm4.71rd 558 . . 3 (𝜑 → (𝜓 ↔ (𝑥𝑉𝜓)))
43eubidv 2585 . 2 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝑥𝑉𝜓)))
5 df-reu 3062 . 2 (∃!𝑥𝑉 𝜓 ↔ ∃!𝑥(𝑥𝑉𝜓))
64, 5syl6rbbr 281 1 (𝜑 → (∃!𝑥𝑉 𝜓 ↔ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wcel 2155  ∃!weu 2581  ∃!wreu 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-mo 2565  df-eu 2582  df-reu 3062
This theorem is referenced by:  frgreu  27506
  Copyright terms: Public domain W3C validator