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

Theorem ralv 2769
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 2473 . 2 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2 vex 2755 . . . 4 𝑥 ∈ V
32a1bi 243 . . 3 (𝜑 ↔ (𝑥 ∈ V → 𝜑))
43albii 1481 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
51, 4bitr4i 187 1 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1362  wcel 2160  wral 2468  Vcvv 2752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-v 2754
This theorem is referenced by:  ralcom4  2774  viin  3961  issref  5026  frecrdg  6427
  Copyright terms: Public domain W3C validator