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Theorem euabsn2 3430
Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
euabsn2 (∃!xφy{xφ} = {y})
Distinct variable groups:   x,y   φ,y
Allowed substitution hint:   φ(x)

Proof of Theorem euabsn2
StepHypRef Expression
1 df-eu 1900 . 2 (∃!xφyx(φx = y))
2 abeq1 2144 . . . 4 ({xφ} = {y} ↔ x(φx {y}))
3 elsn 3382 . . . . . 6 (x {y} ↔ x = y)
43bibi2i 216 . . . . 5 ((φx {y}) ↔ (φx = y))
54albii 1356 . . . 4 (x(φx {y}) ↔ x(φx = y))
62, 5bitri 173 . . 3 ({xφ} = {y} ↔ x(φx = y))
76exbii 1493 . 2 (y{xφ} = {y} ↔ yx(φx = y))
81, 7bitr4i 176 1 (∃!xφy{xφ} = {y})
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240   = 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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-sn 3373
This theorem is referenced by:  euabsn  3431  reusn  3432  absneu  3433  uniintabim  3643  euabex  3952  nfvres  5149  eusvobj2  5441
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