Theorem reuv 3482

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 3368	. 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
2		vex 3458	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 538	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	eubii 2612	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 280	1 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2142  ∃!weu 2595  ∃!wreu 3365  Vcvv 3454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-reu 3368  df-v 3456

This theorem is referenced by:  euen1  9008  updjud  9892  hlimeui  31443