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

Theorem ralv 3192
 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 2901 . 2 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2 vex 3176 . . . 4 𝑥 ∈ V
32a1bi 351 . . 3 (𝜑 ↔ (𝑥 ∈ V → 𝜑))
43albii 1737 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
51, 4bitr4i 266 1 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  ∀wral 2896  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-v 3175 This theorem is referenced by:  ralcom4  3197  viin  4515  issref  5428  ralcom4f  28700  hfext  31460  clsk1independent  37364  ntrneiel2  37404  ntrneik4w  37418
 Copyright terms: Public domain W3C validator