Theorem eueq1 2902

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 2901	. 2 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
3	1, 2	mpbi 144	1 ⊢ ∃!𝑥 𝑥 = 𝐴

Syntax hints:   = wceq 1348  ∃!weu 2019   ∈ wcel 2141  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  eueq2dc  2903  eueq3dc  2904  fsn  5668