Theorem ralv 1867

Description: A universal quantifier restricted to the universe is unrestricted.

Assertion

Ref	Expression
ralv	⊢ (∀x ∈ V φ ↔ ∀xφ)

Proof of Theorem ralv

Step	Hyp	Ref	Expression
1		df-ral 1696	. 2 ⊢ (∀x ∈ V φ ↔ ∀x(x ∈ V → φ))
2		visset 1860	. . . 4 ⊢ x ∈ V
3	2	a1bi 204	. . 3 ⊢ (φ ↔ (x ∈ V → φ))
4	3	albii 1040	. 2 ⊢ (∀xφ ↔ ∀x(x ∈ V → φ))
5	1, 4	bitr4i 183	1 ⊢ (∀x ∈ V φ ↔ ∀xφ)

Syntax hints:   → wi 3   ↔ wb 153  ∀wal 995   ∈ wcel 999  ∀wral 1692  Vcvv 1858

This theorem is referenced by:  ralcom4 1870  viin 2661  ref3w 10566

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-ral 1696  df-v 1859

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