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Theorem reueubd 3419
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 416 . . . 4 (𝜑 → (𝜓𝑥𝑉))
32pm4.71rd 566 . . 3 (𝜑 → (𝜓 ↔ (𝑥𝑉𝜓)))
43eubidv 2673 . 2 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝑥𝑉𝜓)))
5 df-reu 3140 . 2 (∃!𝑥𝑉 𝜓 ↔ ∃!𝑥(𝑥𝑉𝜓))
64, 5syl6rbbr 293 1 (𝜑 → (∃!𝑥𝑉 𝜓 ↔ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  ∃!weu 2654  ∃!wreu 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2624  df-eu 2655  df-reu 3140
This theorem is referenced by:  frgreu  28056
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