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Theorem reuv 2629
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 2360 . 2 (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 2615 . . . 4 𝑥 ∈ V
32biantrur 297 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43eubii 1952 . 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 1434  ∃!weu 1943  ∃!wreu 2355  Vcvv 2612
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-reu 2360  df-v 2614
This theorem is referenced by:  euen1  6449  updjud  6680
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