Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralv Structured version   Visualization version   GIF version

Theorem ralv 3466
 Description: A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
ralv (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)

Proof of Theorem ralv
StepHypRef Expression
1 df-ral 3111 . 2 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2 vex 3444 . . . 4 𝑥 ∈ V
32a1bi 366 . . 3 (𝜑 ↔ (𝑥 ∈ V → 𝜑))
43albii 1821 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
51, 4bitr4i 281 1 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2111  ∀wral 3106  Vcvv 3441 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443 This theorem is referenced by:  ralcom4OLD  3472  viin  4951  ralcom4f  30240  hfext  33757  clsk1independent  40747  ntrneiel2  40787  ntrneik4w  40801
 Copyright terms: Public domain W3C validator