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Theorem reueubd 3303
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 ((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 449 . . . 4 (𝜑 → (𝜓𝑥𝑉))
32pm4.71rd 670 . . 3 (𝜑 → (𝜓 ↔ (𝑥𝑉𝜓)))
43eubidv 2627 . 2 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝑥𝑉𝜓)))
5 df-reu 3057 . 2 (∃!𝑥𝑉 𝜓 ↔ ∃!𝑥(𝑥𝑉𝜓))
64, 5syl6rbbr 279 1 (𝜑 → (∃!𝑥𝑉 𝜓 ↔ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  ∃!weu 2607  ∃!wreu 3052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-nf 1859  df-eu 2611  df-reu 3057
This theorem is referenced by:  frgreu  27422
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