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Theorem euiotaex 4777
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 4772 . . . 4 (x(φx = y) → (℩xφ) = y)
21eqcomd 2027 . . 3 (x(φx = y) → y = (℩xφ))
32eximi 1474 . 2 (yx(φx = y) → y y = (℩xφ))
4 df-eu 1884 . 2 (∃!xφyx(φx = y))
5 isset 2537 . 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 1314  wex 1361   = wceq 1373   wcel 1375  ∃!weu 1881  Vcvv 2533  cio 4759
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2288  df-v 2535  df-sbc 2740  df-un 2900  df-sn 3333  df-pr 3334  df-uni 3533  df-iota 4761
This theorem is referenced by:  iota4an  4780  funfvex  5084
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