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Theorem euabex 3952
 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 3430 . 2 (∃!xφy{xφ} = {y})
2 vex 2554 . . . . 5 y V
3 snexgOLD 3926 . . . . 5 (y V → {y} V)
42, 3ax-mp 7 . . . 4 {y} V
5 eleq1 2097 . . . 4 ({xφ} = {y} → ({xφ} V ↔ {y} V))
64, 5mpbiri 157 . . 3 ({xφ} = {y} → {xφ} V)
76exlimiv 1486 . 2 (y{xφ} = {y} → {xφ} V)
81, 7sylbi 114 1 (∃!xφ → {xφ} V)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  {cab 2023  Vcvv 2551  {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-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918 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-in 2918  df-ss 2925  df-pw 3353  df-sn 3373 This theorem is referenced by: (None)
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