Theorem eueqi 3700

Description: There exists a unique set equal to a given set. Inference associated with euequ 2683. See euequ 2683 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 3699	. 2 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 232	1 ⊢ ∃!𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2114  ∃!weu 2653  Vcvv 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3496

This theorem is referenced by:  eueq2  3701  eueq3  3702  fsn  6897  bj-nuliota  34353  prprval  43696