Theorem reuv 3437

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 3077	. 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 3413	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 534	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	eubii 2604	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 281	1 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)

Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2111  ∃!weu 2587  ∃!wreu 3072  Vcvv 3409

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-reu 3077  df-v 3411

This theorem is referenced by:  euen1  8611  updjud  9409  hlimeui  29135