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Theorem euabex 3933
Description: The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
euabex (∃!xφ → {xφ} V)

Proof of Theorem euabex
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3411 . 2 (∃!xφy{xφ} = {y})
2 vex 2536 . . . . 5 y V
3 snexgOLD 3907 . . . . 5 (y V → {y} V)
42, 3ax-mp 7 . . . 4 {y} V
5 eleq1 2082 . . . 4 ({xφ} = {y} → ({xφ} V ↔ {y} V))
64, 5mpbiri 157 . . 3 ({xφ} = {y} → {xφ} V)
76exlimiv 1471 . 2 (y{xφ} = {y} → {xφ} V)
81, 7sylbi 114 1 (∃!xφ → {xφ} V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228  wex 1362   wcel 1374  ∃!weu 1882  {cab 2008  Vcvv 2533  {csn 3348
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-pow 3899
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-in 2899  df-ss 2906  df-pw 3334  df-sn 3354
This theorem is referenced by: (None)
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