Theorem rexv 3522

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 3146	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 3499	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 533	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	exbii 1848	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 280	1 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1780   ∈ wcel 2114  ∃wrex 3141  Vcvv 3496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-rex 3146  df-v 3498

This theorem is referenced by:  rexcom4OLD  3528  spesbc  3867  exopxfr  5716  elres  5893  elid  6058  dfco2  6100  dfco2a  6101  dffv2  6758  abnex  7481  finacn  9478  ac6s2  9910  ptcmplem3  22664  ustn0  22831  hlimeui  29019  rexcom4f  30236  isrnsiga  31374  prdstotbnd  35074  ac6s3f  35451  moxfr  39296  eldioph2b  39367  kelac1  39670  cbvexsv  40888  sprid  43643