Theorem rexv 3520

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

Step	Hyp	Ref	Expression
1		df-rex 3144	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 3497	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 533	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	exbii 1844	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 280	1 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1776   ∈ wcel 2110  ∃wrex 3139  Vcvv 3494

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-rex 3144  df-v 3496

This theorem is referenced by:  rexcom4OLD  3526  spesbc  3864  exopxfr  5713  elres  5890  elid  6055  dfco2  6097  dfco2a  6098  dffv2  6755  abnex  7478  finacn  9475  ac6s2  9907  ptcmplem3  22661  ustn0  22828  hlimeui  29016  rexcom4f  30233  isrnsiga  31372  prdstotbnd  35071  ac6s3f  35448  moxfr  39287  eldioph2b  39358  kelac1  39661  cbvexsv  40879  sprid  43635