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Theorem rexv 2651
 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 2376 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 2636 . . . 4 𝑥 ∈ V
32biantrur 298 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43exbii 1548 . 2 (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
51, 4bitr4i 186 1 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1433   ∈ wcel 1445  ∃wrex 2371  Vcvv 2633 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-rex 2376  df-v 2635 This theorem is referenced by:  rexcom4  2656  spesbc  2938  abnex  4297  dfco2  4964  dfco2a  4965
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