Theorem rexv 3499

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 3071	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 3478	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 531	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	exbii 1850	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 277	1 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 205   ∧ wa 396  ∃wex 1781   ∈ wcel 2106  ∃wrex 3070  Vcvv 3474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rex 3071  df-v 3476

This theorem is referenced by:  spesbc  3876  exopxfr  5843  elres  6020  elid  6198  dfco2  6244  dfco2a  6245  dffv2  6986  abnex  7746  finacn  10047  ac6s2  10483  ptcmplem3  23565  ustn0  23732  hlimeui  30531  rexcom4f  31748  isrnsiga  33180  prdstotbnd  36754  ac6s3f  37131  moxfr  41518  eldioph2b  41589  kelac1  41893  cbvexsv  43396  sprid  46227