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Theorem riota1a 5609
Description: Property of iota. (Contributed by NM, 23-Aug-2011.)
Assertion
Ref Expression
riota1a ((𝑥𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))

Proof of Theorem riota1a
StepHypRef Expression
1 ibar 295 . 2 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
2 df-reu 2366 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 iota1 4981 . . 3 (∃!𝑥(𝑥𝐴𝜑) → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
42, 3sylbi 119 . 2 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
51, 4sylan9bb 450 1 ((𝑥𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  ∃!weu 1948  ∃!wreu 2361  cio 4965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967
This theorem is referenced by: (None)
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