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Theorem iota1 6045
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 2589 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 sp 2215 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = 𝑧))
3 iotaval 6042 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
43eqeq2d 2775 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝑥 = (℩𝑥𝜑) ↔ 𝑥 = 𝑧))
52, 4bitr4d 273 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = (℩𝑥𝜑)))
6 eqcom 2772 . . . 4 (𝑥 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝑥)
75, 6syl6bb 278 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
87exlimiv 2025 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
91, 8sylbi 208 1 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650   = wceq 1652  wex 1874  ∃!weu 2581  cio 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-v 3352  df-sbc 3597  df-un 3737  df-sn 4335  df-pr 4337  df-uni 4595  df-iota 6031
This theorem is referenced by:  iota2df  6055  sniota  6058  tz6.12-1  6397  opabiota  6450  riota1  6821  riota1a  6822  erovlem  8047  gsumval3lem2  18573  bnj1366  31348  funressndmafv2rn  41971  tz6.12-afv2  41988
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