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Theorem reuv 2638
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 2366 . 2 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 2622 . . . 4 𝑥 ∈ V
32biantrur 297 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43eubii 1957 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
51, 4bitr4i 185 1 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wcel 1438  ∃!weu 1948  ∃!wreu 2361  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-reu 2366  df-v 2621
This theorem is referenced by:  euen1  6499  updjud  6752
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