Theorem eueq 2978

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 2251	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦)
2	1	gen2 1499	. . 3 ⊢ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦)
3	2	biantru 302	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦)))
4		isset 2810	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
5		eqeq1 2238	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
6	5	eu4 2142	. 2 ⊢ (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦)))
7	3, 4, 6	3bitr4i 212	1 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398  ∃wex 1541  ∃!weu 2079   ∈ wcel 2202  Vcvv 2803

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 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2805

This theorem is referenced by:  eueq1  2979  moeq  2982  mosubt  2984  reuhypd  4574  mptfng  5465  gsum0g  13559  gsumval2  13560  upxp  15083