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Theorem reuv 3361
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 3057 . 2 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 3343 . . . 4 𝑥 ∈ V
32biantrur 528 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43eubii 2629 . 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 2139  ∃!weu 2607  ∃!wreu 3052  Vcvv 3340
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-9 2148  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-reu 3057  df-v 3342
This theorem is referenced by:  euen1  8193  updjud  8970  hlimeui  28427
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