Theorem ralv 3516

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 3068	. 2 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2		vex 3492	. . . 4 ⊢ 𝑥 ∈ V
3	2	a1bi 362	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V → 𝜑))
4	3	albii 1817	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
5	1, 4	bitr4i 278	1 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   ∈ wcel 2108  ∀wral 3067  Vcvv 3488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-v 3490

This theorem is referenced by:  viin  5088  ralcom4f  32496  hfext  36147  clsk1independent  44008  ntrneiel2  44048  ntrneik4w  44062