ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralv GIF version

Theorem ralv 2658
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 2380 . 2 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2 vex 2644 . . . 4 𝑥 ∈ V
32a1bi 242 . . 3 (𝜑 ↔ (𝑥 ∈ V → 𝜑))
43albii 1414 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
51, 4bitr4i 186 1 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1297  wcel 1448  wral 2375  Vcvv 2641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-ral 2380  df-v 2643
This theorem is referenced by:  ralcom4  2663  viin  3819  issref  4857  frecrdg  6235
  Copyright terms: Public domain W3C validator