Theorem eueq1 2998

Description: Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)

Hypothesis

Ref	Expression
eueq1.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
eueq1	⊢ ∃!𝑥 𝑥 = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem eueq1

Step	Hyp	Ref	Expression
1		eueq1.1	. 2 ⊢ 𝐴 ∈ V
2		eueq 2997	. 2 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 145	1 ⊢ ∃!𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1402  ∃!weu 2086   ∈ wcel 2209  Vcvv 2821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823

This theorem is referenced by:  eueq2dc  2999  eueq3dc  3000  fsn  5871