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Theorem ralv 3434
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 3087 . 2 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2 vex 3412 . . . 4 𝑥 ∈ V
32a1bi 355 . . 3 (𝜑 ↔ (𝑥 ∈ V → 𝜑))
43albii 1782 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
51, 4bitr4i 270 1 (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1505  wcel 2050  wral 3082  Vcvv 3409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-ral 3087  df-v 3411
This theorem is referenced by:  ralcom4OLD  3440  viin  4848  ralcom4f  30009  hfext  33165  clsk1independent  39759  ntrneiel2  39799  ntrneik4w  39813
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