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Theorem reuv 3464
Description: A unique existential 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 3112 . 2 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 3440 . . . 4 𝑥 ∈ V
32biantrur 531 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43eubii 2630 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
51, 4bitr4i 279 1 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2081  ∃!weu 2611  ∃!wreu 3107  Vcvv 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-reu 3112  df-v 3439
This theorem is referenced by:  euen1  8427  updjud  9209  hlimeui  28708
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