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Theorem rexv 3507
Description: An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
rexv (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)

Proof of Theorem rexv
StepHypRef Expression
1 df-rex 3069 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 3482 . . . 4 𝑥 ∈ V
32biantrur 530 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43exbii 1845 . 2 (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
51, 4bitr4i 278 1 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1776  wcel 2106  wrex 3068  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-v 3480
This theorem is referenced by:  spesbc  3891  exopxfr  5857  elres  6040  elid  6221  dfco2  6267  dfco2a  6268  dffv2  7004  abnex  7776  finacn  10088  ac6s2  10524  ptcmplem3  24078  ustn0  24245  hlimeui  31269  rexcom4f  32497  isrnsiga  34094  prdstotbnd  37781  ac6s3f  38158  moxfr  42680  eldioph2b  42751  kelac1  43052  cbvexsv  44545  sprid  47399
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