Theorem reuv 3357

Description: A uniqueness 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 3053	. 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 3339	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 528	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	eubii 2625	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 267	1 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)

Syntax hints:   ↔ wb 196   ∧ wa 383   ∈ wcel 2135  ∃!weu 2603  ∃!wreu 3048  Vcvv 3336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-12 2192  ax-ext 2736

This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-clab 2743  df-cleq 2749  df-clel 2752  df-reu 3053  df-v 3338

This theorem is referenced by:  euen1  8187  hlimeui  28402