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Theorem iota1 4824
 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 1900 . 2 (∃!xφzx(φx = z))
2 sp 1398 . . . . 5 (x(φx = z) → (φx = z))
3 iotaval 4821 . . . . . 6 (x(φx = z) → (℩xφ) = z)
43eqeq2d 2048 . . . . 5 (x(φx = z) → (x = (℩xφ) ↔ x = z))
52, 4bitr4d 180 . . . 4 (x(φx = z) → (φx = (℩xφ)))
6 eqcom 2039 . . . 4 (x = (℩xφ) ↔ (℩xφ) = x)
75, 6syl6bb 185 . . 3 (x(φx = z) → (φ ↔ (℩xφ) = x))
87exlimiv 1486 . 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 1240   = wceq 1242  ∃wex 1378  ∃!weu 1897  ℩cio 4808 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 3373  df-pr 3374  df-uni 3572  df-iota 4810 This theorem is referenced by:  iota2df  4834  sniota  4837  tz6.12-1  5143  riota1  5429  riota1a  5430  erovlem  6134
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