Theorem ralv 3454

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

Step	Hyp	Ref	Expression
1		df-ral 3070	. 2 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2		vex 3434	. . . 4 ⊢ 𝑥 ∈ V
3	2	a1bi 362	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V → 𝜑))
4	3	albii 1825	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
5	1, 4	bitr4i 277	1 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   ∈ wcel 2109  ∀wral 3065  Vcvv 3430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-v 3432

This theorem is referenced by:  viin  4998  ralcom4f  30797  hfext  34464  clsk1independent  41609  ntrneiel2  41649  ntrneik4w  41663