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Theorem ralv 3191
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 2900 . 2 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2 vex 3175 . . . 4 𝑥 ∈ V
32a1bi 350 . . 3 (𝜑 ↔ (𝑥 ∈ V → 𝜑))
43albii 1736 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
51, 4bitr4i 265 1 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wcel 1976  wral 2895  Vcvv 3172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-ral 2900  df-v 3174
This theorem is referenced by:  ralcom4  3196  viin  4509  issref  5415  ralcom4f  28506  hfext  31266  clsk1independent  37160  ntrneiel2  37200  ntrneik4w  37214
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