ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  euiotaex Structured version   GIF version

Theorem euiotaex 4810
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 4805 . . . 4 (x(φx = y) → (℩xφ) = y)
21eqcomd 2027 . . 3 (x(φx = y) → y = (℩xφ))
32eximi 1473 . 2 (yx(φx = y) → y y = (℩xφ))
4 df-eu 1885 . 2 (∃!xφyx(φx = y))
5 isset 2539 . 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 1226   = wceq 1228  wex 1362   wcel 1374  ∃!weu 1882  Vcvv 2535  cio 4792
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-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
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-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555  df-iota 4794
This theorem is referenced by:  iota4an  4813  funfvex  5117
  Copyright terms: Public domain W3C validator