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Theorem eliotaeu 5200
Description: An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
Assertion
Ref Expression
eliotaeu (𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)

Proof of Theorem eliotaeu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 exsimpr 1618 . 2 (∃𝑦(𝐴𝑦 ∧ ∀𝑥(𝜑𝑥 = 𝑦)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 eliota 5199 . 2 (𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴𝑦 ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
3 df-eu 2029 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
41, 2, 33imtr4i 201 1 (𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351  wex 1492  ∃!weu 2026  wcel 2148  cio 5171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sn 3597  df-uni 3808  df-iota 5173
This theorem is referenced by:  iotam  5203
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