Theorem eueqi 3644

Description: There exists a unique set equal to a given set. Inference associated with euequ 2597. See euequ 2597 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 3643	. 2 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 229	1 ⊢ ∃!𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1539   ∈ wcel 2106  ∃!weu 2568  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434

This theorem is referenced by:  eueq2  3645  eueq3  3646  fsn  7007  bj-nuliota  35228  prprval  44966