Theorem eueqi 3680

Description: There exists a unique set equal to a given set. Inference associated with euequ 2632. See euequ 2632 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 3679	. 2 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 233	1 ⊢ ∃!𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1568   ∈ wcel 2150  ∃!weu 2603  Vcvv 3462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-v 3464

This theorem is referenced by:  eueq2  3681  eueq3  3682  fsn  7135  fineqvnttrclse  35495  bj-nuliota  37641  prprval  48212