Theorem ralv 3456

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 3052	. 2 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2		vex 3433	. . . 4 ⊢ 𝑥 ∈ V
3	2	a1bi 362	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V → 𝜑))
4	3	albii 1821	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
5	1, 4	bitr4i 278	1 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   ∈ wcel 2114  ∀wral 3051  Vcvv 3429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3431

This theorem is referenced by:  viin  5007  ralcom4f  32536  hfext  36365  clsk1independent  44473  ntrneiel2  44513  ntrneik4w  44527