Theorem ralv 2729

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 2440	. 2 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2		vex 2715	. . . 4 ⊢ 𝑥 ∈ V
3	2	a1bi 242	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V → 𝜑))
4	3	albii 1450	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
5	1, 4	bitr4i 186	1 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1333   ∈ wcel 2128  ∀wral 2435  Vcvv 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ral 2440  df-v 2714

This theorem is referenced by:  ralcom4  2734  viin  3908  issref  4968  frecrdg  6355