Theorem reuv 3471

Description: A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)

Assertion

Ref	Expression
reuv	⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)

Proof of Theorem reuv

Step	Hyp	Ref	Expression
1		df-reu 3353	. 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 3446	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 530	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	eubii 2586	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 278	1 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∃!weu 2569  ∃!wreu 3350  Vcvv 3442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-reu 3353  df-v 3444

This theorem is referenced by:  euen1  8976  updjud  9858  hlimeui  31328