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Theorem iota1 4808
Description: Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
iota1 (∃!xφ → (φ ↔ (℩xφ) = x))

Proof of Theorem iota1
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-eu 1885 . 2 (∃!xφzx(φx = z))
2 sp 1382 . . . . 5 (x(φx = z) → (φx = z))
3 iotaval 4805 . . . . . 6 (x(φx = z) → (℩xφ) = z)
43eqeq2d 2033 . . . . 5 (x(φx = z) → (x = (℩xφ) ↔ x = z))
52, 4bitr4d 180 . . . 4 (x(φx = z) → (φx = (℩xφ)))
6 eqcom 2024 . . . 4 (x = (℩xφ) ↔ (℩xφ) = x)
75, 6syl6bb 185 . . 3 (x(φx = z) → (φ ↔ (℩xφ) = x))
87exlimiv 1471 . 2 (zx(φx = z) → (φ ↔ (℩xφ) = x))
91, 8sylbi 114 1 (∃!xφ → (φ ↔ (℩xφ) = x))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1226   = wceq 1228  wex 1362  ∃!weu 1882  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:  iota2df  4818  sniota  4821  tz6.12-1  5125  riota1  5410  riota1a  5411  erovlem  6109
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