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Theorem iota1 5009
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 1952 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 sp 1447 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = 𝑧))
3 iotaval 5006 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
43eqeq2d 2100 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝑥 = (℩𝑥𝜑) ↔ 𝑥 = 𝑧))
52, 4bitr4d 190 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = (℩𝑥𝜑)))
6 eqcom 2091 . . . 4 (𝑥 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝑥)
75, 6syl6bb 195 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
87exlimiv 1535 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
91, 8sylbi 120 1 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1288   = wceq 1290  wex 1427  ∃!weu 1949  cio 4993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2624  df-sbc 2844  df-un 3006  df-sn 3458  df-pr 3459  df-uni 3662  df-iota 4995
This theorem is referenced by:  iota2df  5019  sniota  5022  tz6.12-1  5346  riota1  5642  riota1a  5643  erovlem  6400
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