Theorem reuv 2875

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 2622	. 2 ⊢ (∃!x ∈ V φ ↔ ∃!x(x ∈ V ∧ φ))
2		vex 2863	. . . 4 ⊢ x ∈ V
3	2	biantrur 492	. . 3 ⊢ (φ ↔ (x ∈ V ∧ φ))
4	3	eubii 2213	. 2 ⊢ (∃!xφ ↔ ∃!x(x ∈ V ∧ φ))
5	1, 4	bitr4i 243	1 ⊢ (∃!x ∈ V φ ↔ ∃!xφ)

Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∃!weu 2204  ∃!wreu 2617  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-reu 2622  df-v 2862

This theorem is referenced by: (None)