Theorem ralv 1357

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 1205	. 2 ⊢ (∀x ∈ V φ ↔ ∀x(x ∈ V → φ))
2		visset 1350	. . . 4 ⊢ x ∈ V
3	2	a1bi 172	. . 3 ⊢ (φ ↔ (x ∈ V → φ))
4	3	bial 695	. 2 ⊢ (∀xφ ↔ ∀x(x ∈ V → φ))
5	1, 4	bitr4 154	1 ⊢ (∀x ∈ V φ ↔ ∀xφ)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   ∈ wcel 1092  ∀wral 1201  Vcvv 1348

This theorem is referenced by:  ralcom4 1360

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349

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