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Theorem ralv 2729
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 2440 . 2 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2 vex 2715 . . . 4 𝑥 ∈ V
32a1bi 242 . . 3 (𝜑 ↔ (𝑥 ∈ V → 𝜑))
43albii 1450 . 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 1333  wcel 2128  wral 2435  Vcvv 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ral 2440  df-v 2714
This theorem is referenced by:  ralcom4  2734  viin  3908  issref  4968  frecrdg  6355
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