Theorem rexv 3482

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 3090	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 3459	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 539	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	exbii 1878	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 281	1 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1809   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-v 3457

This theorem is referenced by:  spesbc  3836  exopxfr  5831  elres  6021  elid  6200  dfco2  6248  dfco2a  6249  dffv2  6978  abnex  7757  finacn  10035  ac6s2  10471  ptcmplem3  24192  ustn0  24359  hlimeui  31573  rexcom4f  32796  isrnsiga  34484  onvf1odlem1  35568  prdstotbnd  38426  ac6s3f  38801  moxfr  43406  eldioph2b  43477  kelac1  43773  cbvexsv  45239  sprid  48206