Theorem eueq1 2989

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 2988	. 2 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 145	1 ⊢ ∃!𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1398  ∃!weu 2080   ∈ wcel 2203  Vcvv 2813

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-v 2815

This theorem is referenced by:  eueq2dc  2990  eueq3dc  2991  fsn  5849