Theorem eueqi 3702

Description: There exists a unique set equal to a given set. Inference associated with euequ 2586. See euequ 2586 in the case of a setvar. (Contributed by NM, 5-Apr-1995.)

Hypothesis

Ref	Expression
eueqi.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
eueqi	⊢ ∃!𝑥 𝑥 = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem eueqi

Step	Hyp	Ref	Expression
1		eueqi.1	. 2 ⊢ 𝐴 ∈ V
2		eueq 3701	. 2 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 229	1 ⊢ ∃!𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1534   ∈ wcel 2099  ∃!weu 2557  Vcvv 3469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471

This theorem is referenced by:  eueq2  3703  eueq3  3704  fsn  7138  bj-nuliota  36526  prprval  46826