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Theorem iota1 6540
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 eu6 2572 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 sp 2181 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = 𝑧))
3 iotaval 6534 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
43eqeq2d 2746 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝑥 = (℩𝑥𝜑) ↔ 𝑥 = 𝑧))
52, 4bitr4d 282 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = (℩𝑥𝜑)))
6 eqcom 2742 . . . 4 (𝑥 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝑥)
75, 6bitrdi 287 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
87exlimiv 1928 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
91, 8sylbi 217 1 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wex 1776  ∃!weu 2566  cio 6514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516
This theorem is referenced by:  iota2df  6550  sniota  6554  tz6.12c  6929  tz6.12-1OLD  6931  opabiota  6991  riota1  7409  riota1a  7410  erovlem  8852  gsumval3lem2  19939  bnj1366  34822  funressndmafv2rn  47173  tz6.12-afv2  47190
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