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Theorem iota1 5853
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 2472 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 sp 2051 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = 𝑧))
3 iotaval 5850 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
43eqeq2d 2630 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝑥 = (℩𝑥𝜑) ↔ 𝑥 = 𝑧))
52, 4bitr4d 271 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = (℩𝑥𝜑)))
6 eqcom 2627 . . . 4 (𝑥 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝑥)
75, 6syl6bb 276 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
87exlimiv 1856 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
91, 8sylbi 207 1 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1479   = wceq 1481  wex 1702  ∃!weu 2468  cio 5837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-v 3197  df-sbc 3430  df-un 3572  df-sn 4169  df-pr 4171  df-uni 4428  df-iota 5839
This theorem is referenced by:  iota2df  5863  sniota  5866  tz6.12-1  6197  opabiota  6248  riota1  6614  riota1a  6615  erovlem  7828  gsumval3lem2  18288  bnj1366  30874
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