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Theorem absneu 3433
 Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
Assertion
Ref Expression
absneu ((A 𝑉 {xφ} = {A}) → ∃!xφ)

Proof of Theorem absneu
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sneq 3378 . . . . 5 (y = A → {y} = {A})
21eqeq2d 2048 . . . 4 (y = A → ({xφ} = {y} ↔ {xφ} = {A}))
32spcegv 2635 . . 3 (A 𝑉 → ({xφ} = {A} → y{xφ} = {y}))
43imp 115 . 2 ((A 𝑉 {xφ} = {A}) → y{xφ} = {y})
5 euabsn2 3430 . 2 (∃!xφy{xφ} = {y})
64, 5sylibr 137 1 ((A 𝑉 {xφ} = {A}) → ∃!xφ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  {cab 2023  {csn 3367 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sn 3373 This theorem is referenced by:  rabsneu  3434
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