Theorem eueqi 3696

Description: There exists a unique set equal to a given set. Inference associated with euequ 2682. See euequ 2682 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 3695	. 2 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 232	1 ⊢ ∃!𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1536   ∈ wcel 2113  ∃!weu 2652  Vcvv 3491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-v 3493

This theorem is referenced by:  eueq2  3697  eueq3  3698  fsn  6890  bj-nuliota  34372  prprval  43746