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Theorem absneu 3794
 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.
StepHypRef Expression
1 sneq 3744 . . . . 5 (y = A → {y} = {A})
21eqeq2d 2364 . . . 4 (y = A → ({x φ} = {y} ↔ {x φ} = {A}))
32spcegv 2940 . . 3 (A V → ({x φ} = {A} → y{x φ} = {y}))
43imp 418 . 2 ((A V {x φ} = {A}) → y{x φ} = {y})
5 euabsn2 3791 . 2 (∃!xφy{x φ} = {y})
64, 5sylibr 203 1 ((A V {x φ} = {A}) → ∃!xφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-sn 3741 This theorem is referenced by:  rabsneu  3795
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