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Theorem riota2df 5718
Description: A deduction version of riota2f 5719. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riota2df.1 𝑥𝜑
riota2df.2 (𝜑𝑥𝐵)
riota2df.3 (𝜑 → Ⅎ𝑥𝜒)
riota2df.4 (𝜑𝐵𝐴)
riota2df.5 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
Assertion
Ref Expression
riota2df ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (𝑥𝐴 𝜓) = 𝐵))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)   𝐵(𝑥)

Proof of Theorem riota2df
StepHypRef Expression
1 riota2df.4 . . . 4 (𝜑𝐵𝐴)
21adantr 274 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝐵𝐴)
3 simpr 109 . . . 4 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥𝐴 𝜓)
4 df-reu 2400 . . . 4 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
53, 4sylib 121 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥(𝑥𝐴𝜓))
6 simpr 109 . . . . . 6 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
72adantr 274 . . . . . 6 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝐵𝐴)
86, 7eqeltrd 2194 . . . . 5 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥𝐴)
98biantrurd 303 . . . 4 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ (𝑥𝐴𝜓)))
10 riota2df.5 . . . . 5 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
1110adantlr 468 . . . 4 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓𝜒))
129, 11bitr3d 189 . . 3 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → ((𝑥𝐴𝜓) ↔ 𝜒))
13 riota2df.1 . . . 4 𝑥𝜑
14 nfreu1 2579 . . . 4 𝑥∃!𝑥𝐴 𝜓
1513, 14nfan 1529 . . 3 𝑥(𝜑 ∧ ∃!𝑥𝐴 𝜓)
16 riota2df.3 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
1716adantr 274 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → Ⅎ𝑥𝜒)
18 riota2df.2 . . . 4 (𝜑𝑥𝐵)
1918adantr 274 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝑥𝐵)
202, 5, 12, 15, 17, 19iota2df 5082 . 2 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (℩𝑥(𝑥𝐴𝜓)) = 𝐵))
21 df-riota 5698 . . 3 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
2221eqeq1i 2125 . 2 ((𝑥𝐴 𝜓) = 𝐵 ↔ (℩𝑥(𝑥𝐴𝜓)) = 𝐵)
2320, 22syl6bbr 197 1 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (𝑥𝐴 𝜓) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wnf 1421  wcel 1465  ∃!weu 1977  wnfc 2245  ∃!wreu 2395  cio 5056  crio 5697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-reu 2400  df-v 2662  df-sbc 2883  df-un 3045  df-sn 3503  df-pr 3504  df-uni 3707  df-iota 5058  df-riota 5698
This theorem is referenced by:  riota2f  5719  riota5f  5722
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