Theorem reuv 3481

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 3457	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantrur 540	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
4	3	eubii 2612	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥(𝑥 ∈ V ∧ 𝜑))
5	1, 4	bitr4i 281	1 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  ∃!weu 2595  ∃!wreu 3365  Vcvv 3453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-reu 3368  df-v 3455

This theorem is used by:  euen1  9036  updjud  9942  hlimeui  31707