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Theorem rexv 2781
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 2481 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 2766 . . . 4 𝑥 ∈ V
32biantrur 303 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43exbii 1619 . 2 (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
51, 4bitr4i 187 1 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wex 1506  wcel 2167  wrex 2476  Vcvv 2763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178
This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-rex 2481  df-v 2765
This theorem is referenced by:  rexcom4  2786  spesbc  3075  abnex  4482  dfco2  5169  dfco2a  5170
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