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Theorem euabsn 3431
Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
Assertion
Ref Expression
euabsn (∃!xφx{xφ} = {x})

Proof of Theorem euabsn
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3430 . 2 (∃!xφy{xφ} = {y})
2 nfv 1418 . . 3 y{xφ} = {x}
3 nfab1 2177 . . . 4 x{xφ}
43nfeq1 2184 . . 3 x{xφ} = {y}
5 sneq 3378 . . . 4 (x = y → {x} = {y})
65eqeq2d 2048 . . 3 (x = y → ({xφ} = {x} ↔ {xφ} = {y}))
72, 4, 6cbvex 1636 . 2 (x{xφ} = {x} ↔ y{xφ} = {y})
81, 7bitr4i 176 1 (∃!xφx{xφ} = {x})
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242  wex 1378  ∃!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-sn 3373
This theorem is referenced by:  eusn  3435  args  4637
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