Theorem eueq1 3010

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 3009	. 2 ⊢ (A ∈ V ↔ ∃!x x = A)
3	1, 2	mpbi 199	1 ⊢ ∃!x x = A

Syntax hints:   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  eueq2  3011  eueq3  3012  fnopab2  5209  fsn  5433  composefn  5819