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Theorem eliota 5204
Description: An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)
Assertion
Ref Expression
eliota (𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴𝑦 ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
Distinct variable groups:   𝜑,𝑦   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem eliota
StepHypRef Expression
1 dfiota2 5179 . . 3 (℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
21eleq2i 2244 . 2 (𝐴 ∈ (℩𝑥𝜑) ↔ 𝐴 {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)})
3 eluniab 3821 . 2 (𝐴 {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)} ↔ ∃𝑦(𝐴𝑦 ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
42, 3bitri 184 1 (𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴𝑦 ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wal 1351  wex 1492  wcel 2148  {cab 2163   cuni 3809  cio 5176
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-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sn 3598  df-uni 3810  df-iota 5178
This theorem is referenced by:  eliotaeu  5205
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