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Theorem euiotaex 4825
 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 4820 . . . 4 (x(φx = y) → (℩xφ) = y)
21eqcomd 2042 . . 3 (x(φx = y) → y = (℩xφ))
32eximi 1488 . 2 (yx(φx = y) → y y = (℩xφ))
4 df-eu 1900 . 2 (∃!xφyx(φx = y))
5 isset 2555 . 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 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  Vcvv 2551  ℩cio 4807 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-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 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-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3372  df-pr 3373  df-uni 3571  df-iota 4809 This theorem is referenced by:  iota4an  4828  funfvex  5133
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