Theorem ralv 2873

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 2620	. 2 ⊢ (∀x ∈ V φ ↔ ∀x(x ∈ V → φ))
2		vex 2863	. . . 4 ⊢ x ∈ V
3	2	a1bi 327	. . 3 ⊢ (φ ↔ (x ∈ V → φ))
4	3	albii 1566	. 2 ⊢ (∀xφ ↔ ∀x(x ∈ V → φ))
5	1, 4	bitr4i 243	1 ⊢ (∀x ∈ V φ ↔ ∀xφ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  ∀wral 2615  Vcvv 2860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620  df-v 2862

This theorem is used by:  ralcom4  2878  viin  4026  ssofss  4077