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

Theorem euiotaex 4846
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 4841 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
21eqcomd 2045 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
32eximi 1491 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑))
4 df-eu 1903 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 isset 2558 . 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 1241   = wceq 1243  wex 1381  wcel 1393  ∃!weu 1900  Vcvv 2554  cio 4828
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2309  df-v 2556  df-sbc 2762  df-un 2919  df-sn 3378  df-pr 3379  df-uni 3578  df-iota 4830
This theorem is referenced by:  iota4an  4849  funfvex  5155
  Copyright terms: Public domain W3C validator