Theorem ralv 3192

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 2901	. 2 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
2		vex 3176	. . . 4 ⊢ 𝑥 ∈ V
3	2	a1bi 351	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V → 𝜑))
4	3	albii 1737	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑))
5	1, 4	bitr4i 266	1 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  ∀wral 2896  Vcvv 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-v 3175

This theorem is referenced by:  ralcom4  3197  viin  4515  issref  5428  ralcom4f  28700  hfext  31460  clsk1independent  37364  ntrneiel2  37404  ntrneik4w  37418