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Theorem reuv 3357
Description: A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
Assertion
Ref Expression
reuv (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)

Proof of Theorem reuv
StepHypRef Expression
1 df-reu 3053 . 2 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 3339 . . . 4 𝑥 ∈ V
32biantrur 528 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43eubii 2625 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
51, 4bitr4i 267 1 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2135  ∃!weu 2603  ∃!wreu 3048  Vcvv 3336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-12 2192  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-clab 2743  df-cleq 2749  df-clel 2752  df-reu 3053  df-v 3338
This theorem is referenced by:  euen1  8187  hlimeui  28402
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