Theorem absneu 3795

Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)

Assertion

Ref	Expression
absneu	⊢ ((A ∈ V ∧ {x ∣ φ} = {A}) → ∃!xφ)

Proof of Theorem absneu

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3745	. . . . 5 ⊢ (y = A → {y} = {A})
2	1	eqeq2d 2364	. . . 4 ⊢ (y = A → ({x ∣ φ} = {y} ↔ {x ∣ φ} = {A}))
3	2	spcegv 2941	. . 3 ⊢ (A ∈ V → ({x ∣ φ} = {A} → ∃y{x ∣ φ} = {y}))
4	3	imp 418	. 2 ⊢ ((A ∈ V ∧ {x ∣ φ} = {A}) → ∃y{x ∣ φ} = {y})
5		euabsn2 3792	. 2 ⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y})
6	4, 5	sylibr 203	1 ⊢ ((A ∈ V ∧ {x ∣ φ} = {A}) → ∃!xφ)

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  {csn 3738

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sn 3742

This theorem is referenced by:  rabsneu  3796