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Theorem euabsn2 3413
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 1885 . 2 (∃!xφyx(φx = y))
2 abeq1 2129 . . . 4 ({xφ} = {y} ↔ x(φx {y}))
3 elsn 3365 . . . . . 6 (x {y} ↔ x = y)
43bibi2i 216 . . . . 5 ((φx {y}) ↔ (φx = y))
54albii 1339 . . . 4 (x(φx {y}) ↔ x(φx = y))
62, 5bitri 173 . . 3 ({xφ} = {y} ↔ x(φx = y))
76exbii 1478 . 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 1226   = wceq 1228  wex 1362   wcel 1374  ∃!weu 1882  {cab 2008  {csn 3350
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-sn 3356
This theorem is referenced by:  euabsn  3414  reusn  3415  absneu  3416  uniintabim  3626  euabex  3935  nfvres  5131  eusvobj2  5422
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