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Theorem rexv 2748
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 2454 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
2 vex 2733 . . . 4 𝑥 ∈ V
32biantrur 301 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43exbii 1598 . 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 1485  wcel 2141  wrex 2449  Vcvv 2730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-rex 2454  df-v 2732
This theorem is referenced by:  rexcom4  2753  spesbc  3040  abnex  4432  dfco2  5110  dfco2a  5111
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