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Theorem euiotaex 4781
Description: Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)
Assertion
Ref Expression
euiotaex (∃!xφ → (℩xφ) V)

Proof of Theorem euiotaex
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 iotaval 4776 . . . 4 (x(φx = y) → (℩xφ) = y)
21eqcomd 2026 . . 3 (x(φx = y) → y = (℩xφ))
32eximi 1475 . 2 (yx(φx = y) → y y = (℩xφ))
4 df-eu 1884 . 2 (∃!xφyx(φx = y))
5 isset 2536 . 2 ((℩xφ) V ↔ y y = (℩xφ))
63, 4, 53imtr4i 190 1 (∃!xφ → (℩xφ) V)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1228   = wceq 1230  wex 1364   wcel 1376  ∃!weu 1881  Vcvv 2532  cio 4763
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 1318  ax-7 1319  ax-gen 1320  ax-ie1 1365  ax-ie2 1366  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2003
This theorem depends on definitions:  df-bi 110  df-tru 1233  df-nf 1332  df-sb 1629  df-eu 1884  df-clab 2008  df-cleq 2014  df-clel 2017  df-nfc 2148  df-rex 2287  df-v 2534  df-sbc 2739  df-un 2896  df-sn 3329  df-pr 3330  df-uni 3528  df-iota 4765
This theorem is referenced by:  iota4an  4784  funfvex  5088
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