Theorem eueq 2883

Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
eueq	⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem eueq

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr3 2177	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦)
2	1	gen2 1430	. . 3 ⊢ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦)
3	2	biantru 300	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦)))
4		isset 2718	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
5		eqeq1 2164	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
6	5	eu4 2068	. 2 ⊢ (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦)))
7	3, 4, 6	3bitr4i 211	1 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1333   = wceq 1335  ∃wex 1472  ∃!weu 2006   ∈ wcel 2128  Vcvv 2712

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714

This theorem is referenced by:  eueq1  2884  moeq  2887  mosubt  2889  reuhypd  4429  mptfng  5292  upxp  12632